Energia P de União de Grafos

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Chapter 6
P-energy of join of graphs
In this chapter, we discuss P-energy for the join of two graphs. We further derive the
P-energy for the generalized complements of the join of graphs. We adopt the method
of first finding the characteristic polynomial φP(G) of G in terms of the coronals of the
individual graphs and then determining the value EP(G).
6.1 General results
In this section, we present a generalized expression for the characteristic polynomial of
the graph G1▽G2 and join of k-copies of a graph in terms of the characteristic polynomials
and coronals of the individual graphs. Further we obtain P-energy of join of graphs with
respect to specific vertex partitions.
The M-coronal ΓM(λ ) of G is the sum of elements of the matrix (λ In −M)−1 [31].
Considering the P-matrix AP(G) associated with the vertex partition P of a graph G
we define the P-coronal of G as the sum of elements of the matrix (λ In −AP(G))−1.
It is denoted by ΓAP
(λ ) = 1T
n (λ I −AP(G))−11n, where 1n is a column vector of order
n×1 and 1T
n is its transpose.
In the next theorem, we determine the characteristic polynomial φP(G1▽G2,λ ) of
join of two graphs G1 and G2 in terms of characteristic polynomials φP1(G1,λ ), φP2(G2,λ )
Chapter 6. P-energy of join of graphs
and, P-coronals ΓAP1
(λ ), ΓAP2
(λ ) of G1, G2 corresponding to the vertex partitions P1
and P2 respectively.
Theorem 6.1.1. If P1 and P2 are the vertex partitions of two graphs G1 and G2 of order
n1 and n2 respectively, then
φP(G1▽G2,λ ) = φP1(G1,λ )φP2(G2,λ )[1−ΓAP1
(λ )ΓAP2
(λ )]. (6.1)
Proof. For a vertex partition P = P1 ∪P2 of G1▽G2, the P-matrix
AP(G1▽G2) =
AP1(G1)n1×n1 Jn1×n2
Jn2×n1 AP2(G2)n2×n2
 ,
where Ja×b is the matrix of order a × b whose all elements are 1, for a,b ∈ {n1,n2}.
Therefore, its characteristic polynomial
φP(G1▽G2,λ ) =
∣∣∣∣∣∣∣
λ In1 −AP1(G1)n1×n1 −Jn1×n2
−Jn2×n1 λ In2 −AP2(G2)n2×n2
∣∣∣∣∣∣∣ ,
where Ia is an identity matrix of order a×a for a ∈ {n1,n2}. By Lemma 1.1.6, on simpli-
fying φP(G1▽G2,λ ), we obtain
φP(G1▽G2,λ ) =
∣∣λ In1 −AP1(G1)
∣∣∣∣[λ In2 −AP2(G2)]−B
∣∣. (6.2)
where
B = Jn2×n1[λ In1 −AP1(G1)]
−1Jn1×n2.
148
Chapter 6. P-energy of join of graphs
Now we compute the value of B
B =

1 1 1 · · · 1
1 1 1 · · · 1
1 1 1 · · · 1
...
...
... . . . ...
1 1 1 · · · 1

n2×n1
[λ In1 −AP1(G1)]
−1

1 1 1 · · · 1
1 1 1 · · · 1
1 1 1 · · · 1
...
...
... . . . ...
1 1 1 · · · 1

n1×n2
. (6.3)
Let the matrix be
[λ In1 −AP1(G1)]
−1 =

a11 a12 a13 · · · a1n1
a21 a22 a23 · · · a2n1
a31 a32 a33 · · · a3n1
...
...
... . . . ...
an11 an12 an13 · · · an1n1

n1×n1
. (6.4)
Therefore, by Equation (6.3) and (6.4), further computation gives the value of B.
B =

s s . . . s
s s . . . s
s s . . . s
...
... . . .
...
s s . . . s

n2×n2
where s= s1+s2+ . . .+sn1 and si = ai1+ai2+ . . .+ain1, for i= 1,2, . . . ,n1. The element
s is the sum of all the entries from [λ In1 −AP1(G1)]
−1. By the definition of the P-coronal
149
Chapter 6. P-energy of join of graphs
of a graph, ΓAP1
(λ ) is the sum of all the entries from [λ In1 −AP1(G1)]
−1. Thus,
B =

ΓAP1
(λ ) ΓAP1
(λ ) ΓAP1
(λ ) · · · ΓAP1
(λ )
ΓAP1
(λ ) ΓAP1
(λ ) ΓAP1
(λ ) · · · ΓAP1
(λ )
ΓAP1
(λ ) ΓAP1
(λ ) ΓAP1
(λ ) · · · ΓAP1
(λ )
...
...
... . . . ...
ΓAP1
(λ ) ΓAP1
(λ ) ΓAP1
(λ ) · · · ΓAP1
(λ )

n2×n2
.
Since ΓAP1
(λ ) is a scalar quantity,
B = ΓAP1
(λ )Jn2×n2, (6.5)
where Jn2×n2 is the matrix of order n2×n2 whose all entries are 1. By Equations (6.2) and
(6.5), we obtain
φP(G1▽G2,λ ) = φP1(G1,λ )
∣∣λ In2 −AP2(G2)−ΓAP1
(λ )Jn2×n2
∣∣.
By Lemma 1.1.15,
φP(G1▽G2,λ ) = φP1(G1,λ )
∣∣λ In2 −AP2(G2)
∣∣[1−ΓAP1
(λ )ΓAP2
(λ )
]
.
Hence, the characteristic polynomial of AP(G1▽G2)
φP(G1▽G2,λ ) = φP1(G1,λ )φP2(G2,λ )[1−ΓAP1
(λ )ΓAP2
(λ )].
In the following theorem, we obtain the characteristic polynomial of join G1▽G2 of
two graphs G1 and G2 corresponding to the vertex partition P =V (G1▽G2).
150
Chapter 6. P-energy of join of graphs
Theorem 6.1.2. For the vertex partition P =V (G1▽G2) of G1▽G2,
φP(G1▽G2,λ ) = φP1(A,λ )φP2(B,λ )[1−4ΓA(λ )ΓB(λ )]. (6.6)
where A = AP1(G1)+n2In1 and B = AP2(G2)+n1In2 .
Proof. For the vertex partition P =V (G1▽G2), the P-matrix
AP(G1▽G2) =
AP1(G1)+n2In1 2Jn1×n2
2Jn2×n1 AP2(G2)+n1In2
 .
Therefore, its characteristic polynomial
φP(G1▽G2,λ ) =
∣∣∣∣∣∣∣
λ In1 − [AP1(G1)+n2In1 ] −2Jn1×n2
−2Jn2×n1 λ In2 − [AP2(G2)+n1In2]
∣∣∣∣∣∣∣ ,
By Lemma 1.1.6,
φP(G1▽G2,λ ) =
∣∣λ In1 − [AP1(G1)+n2In1]
∣∣∣∣[λ In2 − [AP2(G2)+n1In2]−C
∣∣. (6.7)
where
C = 4Jn2×n1[λ In2 − [AP1(G1)+n2In1]]
−1Jn1×n2.
Now on computing the value of C as that of B in Theorem 6.1.1, we get
C = 4ΓAP1(G1)+n2In1
(λ )Jn2×n2. (6.8)
Let A = AP1(G1)+ n2In1 and B = AP2(G2)+ n1In2 . Thus, by Equations (6.7) and (6.8)
we obtain
φP(G1▽G2,λ ) = φP1(A,λ )
∣∣λ In2 −B−4ΓA(λ )Jn2×n2
∣∣.
151
Chapter 6. P-energy of join of graphs
By Lemma 1.1.15,
φP(G1▽G2,λ ) = φP1(A,λ )
∣∣λ In2 −B
∣∣[1−4ΓA(λ )ΓB(λ )
]
.
Hence,
φP(G1▽G2,λ ) = φP1(A,λ )φP2(B,λ )[1−4ΓA(λ )ΓB(λ )].
It has been observed that the adjacency matrix of a regular graph has a constant row
sum [10, 31, 32], but it is worthy to note that in case of P-matrix, not only regular graphs
but non-regular graphs can also have P-matrices with constant row sum. Note that, row
sum is the sum of all the elements in a row of the given matrix.
We now attempt to determine the exact value of EP(G1▽G2) in the special case when
AP(G1) and AP(G2) has a constant row sums R1 and R2 respectively. In order to do so,
first we prove the following lemma that gives the value of ΓAP
(λ ).
Lemma 6.1.3. Let G be a graph of order n such that its P-matrix AP(G) corresponding
to the partition P has a constant row sum R. Then
ΓAP
(λ ) =
n
λ −R
. (6.9)
Proof. Let AP(G) be P-matrix of G and R be its constant row sum. Thus,
AP(G)1n =

a11 a12 a13 · · · a1n1
a21 a22 a23 · · · a2n1
a31 a32 a33 · · · a3n1
...
...
... . . . ...
an11 an12 an13 · · · an1n1

1n
152
Chapter 6. P-energy of join of graphs
AP(G)1n =

a11 +a12 +a13 + · · ·+a1n1
a21 +a22 +a23 + · · ·+a2n1
a31 +a32 +a33 + · · ·+a3n1
...
an11 +an12 +an13 + · · ·+an1n1

AP(G)1n =R1n, where 1n is a column vector of order n×1.
Therefore, by the definition of an eigenvalue, R is an eigenvalue of AP(G).
[λ In −AP(G)]1n = (λ −R)1n.
Thus,
1T
n
1
(λ −R)
1n =1T
n
1
[λ In −AP(G)]
1n
=1T
n [λ In −AP(G)]−11n, (6.10)
where 1T
n is the transpose of 1n and by the definition of P-coronal of a graph,
ΓAP
(λ ) = 1T
n [λ In −AP(G)]−11n. (6.11)
Hence, by Equation (6.10) and (6.11)
ΓAP
(λ ) =
1T
n 1n
λ −R
=
n
λ −R
.
Using Theorem 6.1.1 and Lemma 6.1.3, we derive the characteristic polynomial of P-
matrix of join of two graphs G1 and G2 in the special case when P-matrices of G1 and G2
153
Chapter 6. P-energy of join of graphs
have a constant row sum R1 and R2 respectively. By row sum, we mean sum of the ele-
ments of a row from a matrix.
Lemma 6.1.4. If G1 and G2 are graphs of order n1 and n2 having vertex partitions P1
and P2 respectively such that AP1(G1) has a constant row sum R1 and AP2(G2) has a
constant row sum R2, then with respect to the vertex partition P = P1 ∪P2 of G1▽G2,
φP(G1▽G2,λ ) = φP1(G1,λ )φP2(G2,λ )
[
(λ −R1)(λ −R2)−n1n2
(λ −R1)(λ −R2)
]
. (6.12)
Proof. By Lemma 6.1.3, for the vertex partition P1
ΓAP1
(λ ) =
n1
λ −R1
(6.13)
and for the vertex partition P2
ΓAP2
(λ ) =
n2
λ −R2
. (6.14)
Therefore, by substituting Equations (6.13) and (6.14) in Equation (6.1)
φP(G1▽G2,λ ) = φP1(G1,λ )φP2(G2,λ )
[
1− n
λ −R1
n
λ −R2
]
= φP1(G1,λ )φP2(G2,λ )
[
(λ −R1)(λ −R2)−n1n2
(λ −R1)(λ −R2)
]
.
In the following theorem, we derive the P-energy of join of two graphs G1 and G2
in the special case where their corresponding P-matrices AP1(G1) and AP2(G2) have
constant row sum.
154
Chapter6. P-energy of join of graphs
Theorem 6.1.5. Let G1 and G2 be two graph of order n1 and n2 respectively. Let P1
and P2 be their respective vertex partitions. If AP1(G1) and AP2(G2) have constant row
sum, then for a vertex partition P = P1 ∪P2 of G1▽G2
EP(G1▽G2) = EP1(G1)+EP2(G2)− ∑
i=1,2
|Ri|
+
1
2
{∣∣∣∣(R1 +R2)+
√
(R1 +R2)2 −4(R1R2 −n1n2)
∣∣∣∣
+
∣∣∣∣(R1 +R2)−
√
(R1 +R2)2 −4(R1R2 −n1n2)
∣∣∣∣}
where Ri is a constant row sum of AP(Gi), for i = 1,2.
Proof. By Lemma 6.1.4, we have
φP(G1▽G2,λ ) = φP1(G1,λ )φP2(G2,λ )
[
(λ −R1)(λ −R2)−n1n2
(λ −R1)(λ −R2)
]
(6.15)
where P1 and P2 are the vertex partitions of G1 and G2. Therefore, Equation (6.15) can
be written as
[λ −R1][λ −R2]φP(G1▽G2,λ ) =φP1(G1,λ )φP2(G2,λ )[(λ −R1)(λ −R2)−n1n2]
=φP1(G1,λ )φP2(G2,λ )[λ
2 − (R1 +R2)λ +(R1R2 −n1n2)].
(6.16)
Let the left hand side of the Equation (6.16) be L(λ ) and the right hand side be R(λ ).
Thus, the roots of the equations L(λ ) = 0 and R(λ ) = 0 are same. Therefore, the sum of
the absolute values of the roots of these equations are also same. Thus,
∑
i=1,2
|Ri|+EP(G1▽G2) = EP1(G1)+EP2(G2)
+
1
2
{∣∣∣∣(R1 +R2)+
√
(R1 +R2)2 −4(R1R2 −n1n2)
∣∣∣∣
+
∣∣∣∣(R1 +R2)−
√
(R1 +R2)2 −4(R1R2 −n1n2)
∣∣∣∣}.
155
Chapter 6. P-energy of join of graphs
Hence,
EP(G1▽G2) =EP1(G1)+EP2(G2)− ∑
i=1,2
|Ri|
+
1
2
{∣∣∣∣(R1 +R2)+
√
(R1 +R2)2 −4(R1R2 −n1n2)
∣∣∣∣
+
∣∣∣∣(R1 +R2)−
√
(R1 +R2)2 −4(R1R2 −n1n2)
∣∣∣∣}.
Recall that, the robust P-energy of a graph G is a P-energy of a graph G for P =
V (G). In the next theorem, we determine robust P-energy of G1▽G2.
Theorem 6.1.6. The robust P-energy of G1▽G2
EPr(G1▽G2) =
n1
∑
i=1
|λi +n2|+
n2
∑
i=1
|λ ′
i +n1|− ∑
i=1,2
|R′
i|
+
1
2
{∣∣∣∣(R′
1 +R′
2)+
√
(R′
1 +R′
2)
2 −4(R′
1R′
2 −4n1n2)
∣∣∣∣
+
∣∣∣∣(R′
1 +R′
2)−
√
(R′
1 +R′
2)
2 −4(R′
1R′
2 −4n1n2)
∣∣∣∣}
where λi, λ ′
i are the eigenvalues of AP1(G1), AP2(G2) respectively and Ri is a constant
row sum of AP(Gi), for i = 1,2.
Proof. By Theorem 6.1.2, for the vertex partition P =V (G1▽G2) of G1▽G2,
φP(G1▽G2,λ ) = φP1(A,λ )φP2(B,λ )[1−4ΓA(λ )ΓB(λ )]. (6.17)
where A = AP1(G1)+ n2In1 and B = AP2(G2)+ n1In2 . By the method used in Lemma
6.1.4, we compute Equation (6.17) and obtain the following equation:
φP(G1▽G2,λ ) = φP1(A,λ )φP2(B,λ )
[
(λ −R′
1)(λ −R′
2)−4n1n2
(λ −R′
1)(λ −R′
2)
]
(6.18)
156
Chapter 6. P-energy of join of graphs
where R′
1 = R1 + n2 and R′
2 = R2 + n1, A = AP1(G1)+ n2In1 and B = AP2(G2)+ n1In2 .
By similar procedure as adopted in Theorem 6.1.5, we get the following equation
[λ −R′
1][λ −R′
2]φP(G1▽G2,λ ) =φP1(A,λ )φP2(B,λ )[(λ −R′
1)(λ −R′
2)−4n1n2]
=φP1(A,λ )φP2(B,λ )[λ
2 − (R′
1 +R′
2)λ +(R′
1R′
2 −4n1n2)].
(6.19)
Let the left hand side of the Equation (6.19) be L(λ ) and the right hand side be R(λ ).
Thus, the roots of the equations L(λ ) = 0 and R(λ ) = 0 are same. Therefore, the sum of
the absolute values of the roots of these equations are also same. Thus,
∑
i=1,2
|R′
i|+EP(G1▽G2) = EP1(A)+EP2(B)
+
1
2
{∣∣∣∣(R′
1 +R′
2)+
√
(R′
1 +R′
2)
2 −4(R′
1R′
2 −4n1n2)
∣∣∣∣
+
∣∣∣∣(R′
1 +R′
2)−
√
(R′
1 +R′
2)
2 −4(R′
1R′
2 −4n1n2)
∣∣∣∣}. (6.20)
Let S be a square matrix and λ be its eigenvalue, then Sx = λx, where x is an eigenvector.
Thus,
(S+aI)x = Sx+aIx = λx+ax = (λ +a)x,
where a is a scalar quantity and I be an identity matrix. That is, if λ is an eigenvalue of
a square matrix S, then λ + a is an eigenvalue of the square matrix S+ aI. Similarly,
if λ1,λ2,λ3, . . . ,λn1 and λ ′
1,λ
′
2,λ
′
3, . . . ,λ
′
n2
are the eigenvalues of AP1(G1) and AP2(G2)
respectively, then λ1 + n2,λ2 + n2,λ3 + n2, . . .+ n2,λn1 + n2 and λ ′
1 + n1,λ
′
2 + n1,λ
′
3 +
n1, . . . ,λ
′
n2
+ n1 are the eigenvalues of A = AP1(G1) + n2In1 and B = AP2(G2) + n1In2
respectively. Therefore, the sum of the absolute values of the eigenvalues of A and B can
157
Chapter 6. P-energy of join of graphs
be written as
EP1(A) =
n1
∑
i=1
|λi +n2| and EP2(B) =
n2
∑
i=1
|λ ′
i +n1|. (6.21)
Hence, by Equations (6.20) and (6.21)
EP(G1▽G2) =
n1
∑
i=1
|λi +n2|+
n2
∑
i=1
|λ ′
i +n1|− ∑
i=1,2
|R′
i|
+
1
2
{∣∣∣∣(R′
1 +R′
2)+
√
(R′
1 +R2)2 −4(R′
1R′
2 −4n1n2)
∣∣∣∣
+
∣∣∣∣(R′
1 +R′
2)−
√
(R′
1 +R′
2)
2 −4(R′
1R′
2 −4n1n2)
∣∣∣∣}.
The join of k-copies of a graph F is obtained by joining every vertex of F to every
vertex of the remaining (k−1) copies of F , where k is any positive number greater than
or equal to two. In the next theorem, we derive an expression for the characteristic poly-
nomial for the join of k-copies of a graph F in terms of characteristic polynomial of F .
Further, using this expression we obtain EP(G) where G is the join of k-copies of F when
AP(F) has constant row sum. It is to be noted that, Ci denotes the ith column of a matrix
for i = 1,2,3, . . . ,n.
Theorem 6.1.7. Let F be a graph of order t and P1 be its vertex partition. If G is the join
of k-copies of F, then for the vertex partition P of G having k elements, each of which is
a P1,
φP(G,λ ) =
[
φP1(F,λ )
]k[1− (k−1)ΓAP1
(λ )
][
1+ΓAP1
(λ )
](k−1)
. (6.22)
158
Chapter 6. P-energy of join of graphs
Proof. By the choice of P , we have the P-matrix of G.
AP(G) =

AP1(F) J J . . . J
J AP1(F) J . . . J
J J AP1(F) . . . J
...
...
... . . . ...
J J J . . . AP1(F)

k×k
.
Therefore, the characteristic polynomial of AP(G)
φP(G,λ ) =
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
λ I −AP1(F) −J −J . . . −J
−J λ I −AP1(F) −J . . . −J
−J −J λ I −AP1(F) . . . −J
...
...
...
. . .
...
−J −J −J . . . λ I −AP1(F)
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
k×k
.
Now we apply some row and column operations on φP(G,λ ), add
k
∑
i=2
Ci to the first
column
φP(G,λ ) =
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
λ I −AP1(F)− (k−1)J −J −J . . . −J
λ I −AP1(F)− (k−1)J λ I −AP1(F) −J . . . −J
λ I −AP1(F)− (k−1)J −J λ I −AP1(F) . . . −J
...
...
...
. . .
...
λ I −AP1(F)− (k−1)J −J −J . . . λ I −AP1(F)
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
k×k
.
159
Chapter 6. P-energy of join of graphs
Then, subtract first row from ith row, for i = 2,3, . . . ,k
φP(G,λ ) =
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
a −J −J . . . −J
0 b 0 . . . 0
0 0 b . . . 0
...
...
... . . . ...
0 0 0 . . . b
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
k×k
.
where a = λ I −AP1(F)− (k− 1)J and b = λ I −AP1(F)+ J. Since φP(G,λ ) is trian-
gular,
φP(G,λ ) =
∣∣λ I −AP1(F)− (k−1)J
]
|
∣∣λ I −AP1(F)+ J
∣∣(k−1)
.
By Lemma 1.1.15,
φP(G,λ ) =
[
1− (k−1)ΓAP1
(λ )
]∣∣λ I −AP1(F)
∣∣{[1+ΓAP1
(λ )]
∣∣λ I −AP1(F)
∣∣}(k−1)
=
[
φP1(F,λ )
]k[1− (k−1)ΓAP1
(λ )
][
1+ΓAP1
(λ )
](k−1)
.
In the following theorem, we consider graphs G and F as described in Theorem 6.1.7.
The vertex partition of G and F are P and P1 respectively. Note that, AP1(F) is of order
t × t and It is an identity matrix of order t × t.
Theorem 6.1.8. For P =V (G),
φP(G,λ ) =
[
φP1(C,λ )
]k[1−2(k−1)ΓC(λ )
][
1+2ΓC(λ )
](k−1)
where C = AP1(F)+(n− t)It .
160
Chapter 6. P-energy of join of graphs
Proof. For P =V (G), we have the P-matrix of G,
AP(G) =

C 2J 2J . . . 2J
2J C 2J . . . 2J
2J 2J C . . . 2J
...
...
... . . . ...
2J 2J 2J . . . C

k×k
where C = AP1(F)+(n− t)It . Therefore, the characteristic polynomial of AP(G)
φP(G,λ ) =
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
λ I −C −2J −2J . . . −2J
−2J λ I −C −2J . . . −2J
−2J −2J λ I −C . . . −2J
...
...
... . . . ...
−2J −2J −2J . . . λ I −C
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
.
Add
k
∑
i=2
Ci to the first column
φP(G,λ ) =
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
λ I −C−2(k−1)J −2J −2J . . . −2J
λ I −C−2(k−1)J λ I −C −2J . . . −2J
λ I −C−2(k−1)J −2J λ I −C . . . −2J
...
...
... . . . ...
λ I −C−2(k−1)J −2J −2J . . . λ I −C
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
k×k
.
161
Chapter 6. P-energy of join of graphs
Then, subtract the first row from ith row, for i = 2,3, . . . ,k
φP(G,λ ) =
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
λ I −C−2(k−1)J −2J −2J . . . −2J
0 λ I −C+2J 0 . . . 0
0 0 λ I −C+2J . . . 0
...
...
... . . . ...
0 0 0 . . . λ I −C+2J
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
k×k
.
Therefore, on evaluating the determinant, we obtain
φP(G,λ ) =
∣∣λ I −C−2(k−1)J
∣∣∣∣λ I −C+2J
∣∣(k−1)
.
By Lemma 1.1.15,
φP(G,λ ) =
[
1−2(k−1)ΓC(λ)
]∣∣λ I −C
∣∣{[1+2ΓC(λ )]
∣∣λ I −C
∣∣}(k−1)
=(|λ I −C|)k[1−2(k−1)ΓC(λ )][1+2ΓC(λ )]
(k−1)
=
[
φP1(C,λ )
]k[1−2(k−1)ΓC(λ )
][
1+2ΓC(λ )
](k−1)
.
where C = AP1(F)+(n− t)It .
In the succeeding theorem, we consider the join G of k-copies of a graph F wherein
the P-matrix of F has a constant row sum and obtain the corresponding P-energy of G.
Theorem 6.1.9. Let F be a graph of order t and G be the join of k-copies of F. Let
P1 and P be the vertex partitions of G and F respectively. If AP1(F) has a constant
row sum R, then for a vertex partition P such that it is the union of k-copies of P1,
EP(G) = kEP1(F)− k|R|+ |R+(k−1)t|+(k−1)|R− t|. (6.23)
162
Chapter 6. P-energy of join of graphs
Proof. By Lemma 6.1.3, we have
ΓAP1
(λ ) =
t
λ −R
(6.24)
and by Theorem 6.1.7,
φP(G,λ ) =
[
φP1(F,λ )
]k[1− (k−1)ΓAP1
(λ )
][
1+ΓAP1
(λ )
](k−1)
. (6.25)
Therefore, by Equations (6.24) and (6.25) we obtain
φP(G,λ ) =
[
φP1(F,λ )
]k[1− (k−1)
t
λ −R
][
1+
t
λ −R
](k−1)
=
[
φP1(F,λ )
]k
[
λ −R− (k−1)t
λ −R
][
λ −R+ t
λ −R
](k−1)
φP(G,λ ) =
[
φP1(F)
λ −R
]k[
λ −R− (k−1)t
][
λ −R+ t
](k−1)
. (6.26)
Thus, Equation (6.26) can be written as
(λ −R)k
φP(G,λ ) =
[
φP1(F)
]k[
λ −R− (k−1)t
][
λ −R+ t
](k−1)
. (6.27)
Consider the left hand side and the right hand side of the Equation (6.27) as S1(λ ) and
S2(λ ) respectively. The roots of equation S1(λ ) = 0 and S2(λ ) = 0 are same. Therefore,
the sum of the absolute values of their roots are also same. Thus,
k|R|+EP(G) = kEP1(F)+ |R+(k−1)t|+(k−1)|R− t|. (6.28)
On simplifying Equation (6.28), we obtain the required result.
163
Chapter 6. P-energy of join of graphs
Theorem 6.1.10. The robust P-energy of join G of k-copies of a graph F for the vertex
partition Pr =V (G) is
EPr(G) = k
t
∑
i=1
|λi +n− t|− k|R′|+ |R′+2(k−1)t|+(k−1)|R′−2t|
where λi is an eigenvalue of AP1(F) for i = 1,2, . . . , t and R′ is the constant row sum of
AP1(F)+(n− t)It .
Proof. By Theorem 6.1.8, for P =V (G),
φP(G,λ ) =
[
φP1(C,λ )
]k[1−2(k−1)ΓC(λ )
][
1+2ΓC(λ )
](k−1) (6.29)
where C = AP1(F)+(n− t)It and by Lemma 6.1.3, we have
ΓAP1
(λ ) =
t
λ −R′ (6.30)
where R′ = R+(n− t). Therefore, using Equation (6.29), Equation (6.30) can be written
as follows:
φP(G,λ ) =
[
φP1(C,λ )
]k[1−2(k−1)ΓC(λ )
][
1+2ΓC(λ )
](k−1)
=
[
φP1(C,λ )
]k[1− 2(k−1)t
λ −R′
][
1+
2t
λ −R′
](k−1)
=
[
φP1(C,λ )
λ −R′
]k
[λ − (R′+2(k−1)t)][λ − (R′−2t)](k−1)
(λ −R′)k
φP(G,λ ) =
[
φP1(C,λ )
]k
[λ − (R′+2(k−1)t)][λ − (R′−2t)](k−1). (6.31)
Consider the left hand side and the right hand side of the Equation (6.31) as S1(λ ) and
S2(λ ) respectively. The roots of equation S1(λ ) = 0 and S2(λ ) = 0 are same. Therefore,
the sum of the absolute values of their roots are also same.
164
Chapter 6. P-energy of join of graphs
Thus,
k|R′|+EP(G) = kEP1(F)+ |R
′
+2(k−1)t|+(k−1)|R
′
−2t|
Therefore,
EP(G) = kEP1(F)− k|R′|+ |R
′
+2(k−1)t|+(k−1)|R
′
−2t|
where R′ = R+(n− t) and R is a constant row sum of AP1(F).
6.2 Generalized complements of join of graphs
We begin this section with some results for P-energy of complement of join of graphs,
followed by results for the P-energy of generalized complements of join of graphs.
The following theorem provides the P-energy of complement of join of k-copies of
a graph:
Theorem 6.2.1. Let G be a graph of order n obtained by the join of k-copies of a graph
F of order t and P be its vertex partition such that it is the union of k-copies of vertex
partition P1 of F. If G is the complement of G, then for the vertex partition P of G,
EP(G) =
k
∑
i=1
EPi(F).
Proof. Let G be the join of k-copies of F and G be its complement. Let P be the vertex
partition of G such that it is the union of k-copies of P1 where P1 is the vertex partition
of F . We consider the same vertex partition P for G also. Therefore, for a vertex partition
165
Chapter 6. P-energy of join of graphs
P such that it is the union of k-copies of P1, the P-matrix of G is
AP(G) =

AP1(F) 0 0 . . . 0
0 AP2(F) 0 . . . 0
0 0 AP3(F) . . . 0
...
...
... . . . ...
0 0 0 . . . APk(F)

k×k
(6.32)
where 0’s are the block zero matrices of order n1 ×n1. Therefore, by Lemma 1.1.6 and
Equation (6.32) we get
φP(G,λ ) = φP1(F ,λ )φP2(F ,λ ) . . .φPk(F ,λ ). (6.33)
Let left hand side of the Equation (6.33) be L(λ ) and right hand side be R(λ ). Thus, the
roots of the equations L(λ ) = 0 and R(λ ) = 0 are same. So, the sum of the absolute values
of the roots of these equations are also same. Therefore,
EP(G) = EP1(F)+EP2(F)+ . . .+EPk(F).
In the next theorem, we determine the characteristic polynomial of complement of the
join of k-copies of F . We omit its proof because it is similar to that of Theorem 6.1.7.
Lemma 6.2.2. Let F be a graph of order t and P1 be its vertex partition such that
P1 =V (F). Let G be a graph of order n obtained by the join of k-copies of F and P be
its vertex partition. Then for P =V (G),
φP(G,λ ) =
[
φP1(D,λ )
]k[1+(k−1)ΓD(λ )
][
1+ΓD(λ )
](k−1)
where D = AP1(F)+(n− t)It .
166
Chapter 6. P-energy of join of graphs
In the following theorem, we consider a graph F whose P-matrix has a constant row
sum and find the P-energy of complement of join of k-copies of F using Lemmas 6.1.3
and 6.2.2.
Theorem 6.2.3. Let G be a graph of order n obtained by the join of k-copies of a graph
F of order t such that P and P1 are their vertex partitions respectively. Let G be the
complement of G. If AP1(F) has constant row sum R, then for P =V (G)
EP(G) = k
t
∑
i=1
|λi +n− t|− k|R′|+ |R′− (k−1)t|+(k−1)|R′+ t|. (6.34)
Proof. Let G be the join of k-copies of F and G be the complement of G. Let P be the
vertex partition of G and we retain the same partition P in G also. For the vertex partition
P =V (G), by Lemma 6.2.2, the characteristic polynomial of AP(G) is
φP(G,λ ) =
[
φP1(D,λ )
]k[1+(k−1)ΓP1(D)
][
1+ΓP1(D)
](k−1)
, (6.35)
where D = AP1(F)+(n− t)It . Let AP1(F) be a P-matrix of F which has a constant row
sum R. Then AP1(F)+ (n− t)It has constant row sum R′ = R+n− t. Thus, by Lemma
6.1.3
ΓP1(F) =
t
λ −R′ . (6.36)
Therefore, by Equations (6.35) and (6.36)
φP(G,λ ) =
[
φP1(D,λ )
]k[1+(k−1)
t
λ −R′
][
1+
t
λ −R′
](k−1)
φP(G,λ ) =
[
φP1(D)
λ −R′
]k[
λ −R′+(k−1)t
][
λ −R′+ t
](k−1)
(λ −R′)k
φP(G,λ ) =
[
φP1(D)
]k[
λ −R′+(k−1)t
][
λ −R′+ t
](k−1)
. (6.37)
Let left hand side of the Equation (6.37) be L(λ ) and right hand side be R(λ ). Thus,
the roots of equations L(λ ) = 0 and R(λ ) = 0 are same. So, the sum of the absolute values
167
Chapter 6. P-energy of join of graphs
of the roots of these equations are also same. Therefore,
k|R′|+EP(G) = k
t
∑
i=1
|λi +n− t|+ |R′− (k−1)t|+(k−1)|R′− t|
EP(G) = k
t
∑
i=1
|λi +n− t|− k|R′|+ |R′− (k−1)t|+(k−1)|R′− t|,
where λi is an eigenvalue of D = AP1(F)+ (n− t)It , R′ = R+ n− t and R is a constant
row sum of AP1(F).
The following theorem gives the P-energy of complement of join of k graphs G1,G2,
G3, . . . ,Gk in terms of P-energy of component graphs with corresponding vertex parti-
tions. It is to be noted that G1,G2, . . . ,Gk may not be the same.
Theorem 6.2.4. Let G be the join of k graphs G1,G2, . . . ,Gk such that P1,P2, . . . ,Pk
are the vertex partitions of G1,G2, . . . ,Gk respectively and P =
k⋃
i=1
Pi be the vertex
partition of G. If G is the complement of G, then for the vertex partition P =
k⋃
i=1
Pi of
V (G),
EP(G) = EP1(G1)+EP2(G2)+ . . .+EPk(Gk).
Proof. The P-matrix of G is
AP(G) =

AP1(G1) 0n1×n2 0n1×n3 . . . 0n1×nn
0n2×n1 AP2(G2) 0n2×n3 . . . 0n2×nn
0n3×n1 0n3×n2 AP3(G3) . . . 0n3×nn
...
...
... . . . ...
0nn×n1 0nn×n2 0nn×n3 . . . APk(Gk)

k×k
(6.38)
where APi(Gi) is the P-matrix of Gi with respect to the vertex partition Pi, for i =
1,2, . . . ,k and 0’s are the block zero matrices. Therefore, by Lemma 1.1.6 and Equation
168
Chapter 6. P-energy of join of graphs
(6.38)
φP(G,λ ) = φP1(G1,λ )φP2(G2,λ ) . . .φPk(Gk,λ ). (6.39)
Let left hand side ofthe Equation (6.39) be L(λ ) and right hand side be R(λ ). Let R(λ ) =
R1(λ )R2(λ )R3(λ ) . . .Rk(λ ). Thus, sum of the absolute values of eigenvalues of R(λ ) is
the same as the sum of the absolute values eigenvalues of R1(λ )R2(λ )R3(λ ) . . .Rk(λ ).
Therefore, the roots of the equations L(λ ) = 0 and R(λ ) = 0 are same. So, the sum of the
absolute values of the roots of these equations are also same. Therefore,
n
∑
i=1
|λi|=
n1
∑
i=1
|λi|+
n2
∑
i=1
|λi|+
n3
∑
i=1
|λi|+ . . .+
nk
∑
i=1
|λi|.
Hence,
EP(G) = EP1(G1)+EP2(G2)+ . . .+EPk(Gk).
In the next theorem, we determine the P-energy of k-complement of join of k graphs
G1,G2, . . . ,Gk in terms of P-energy of k-complements of G1,G2, . . . ,Gk. Note that, these
k graphs may not be the same.
Theorem 6.2.5. For a graph G = G1 ▽G2 ▽ . . .▽Gk of order n such that P =
k
∑
i=1
Pi ,
EP(G)k =
k
∑
i=1
EPi(Gi)k. (6.40)
where (Gi)k is the k-complement of the graph Gi and Pi is the vertex partition of Gi, for
i = 1,2, . . . ,k.
169
Chapter 6. P-energy of join of graphs
Proof. Let G = G1 ▽G2 ▽ . . .▽Gk be a graph of order n and P =
k
∑
i=1
Pi. Therefore,
the P-matrix of (G)k is
AP(G)k =

AP1(G1)k 0n1×n2 0n1×n3 . . . 0n1×nn
0n2×n1 AP2(G2)k 0n2×n3 . . . 0n2×nn
0n3×n1 0n3×n2 AP3(G3)k . . . 0n3×nn
...
...
... . . . ...
0nn×n1 0nn×n2 0nn×n3 . . . APk(Gk)k

k×k
(6.41)
where AP1(G1)k,AP2(G2)k, . . . ,APk(Gk)k are the P-matrix of (Gi)k with respect to the
vertex partition Pi, for i = 1,2, . . . ,k and 0’s are the block zero matrices. Therefore, by
Lemma 1.1.6 and Equation (6.41),
φP((G)k,λ ) = φP1((G1)k,λ )φP2((G2)k,λ ) . . .φPk((Gk)k,λ ).
Thus, the P-energy of (G)k is
EP(G)k = EP1(G1)k +EP2(G2)k + . . .+EPk(Gk)k.
In the subsequent theorem, we determine the characteristic polynomial of k(i)-complement
of join of k-copies of a graph. We omit its proof, since the process of obtaining Equation
(6.42) is similar to that of Equation (6.22) in the proof of Theorem 6.1.7.
Theorem 6.2.6. Let F be a graph of order t and let P1 be its vertex partition. If G is a
graph of order n obtained by the join of k-copies of F such that P is its vertex partition,
170
Chapter 6. P-energy of join of graphs
then for P of (G)k(i) such that it is the union of k-copies of P1,
φP((G)k(i),λ ) =
[
φP1((F)k(i),λ )
]k[1− (k−1)ΓAP1
(λ )
][
1+ΓAP1
(λ )
](k−1)
. (6.42)
By considering the case when AP1(F) has constant row sum R, in the next theorem
we obtain the the corresponding P-energy by the similar method as shown in the proof
of Theorem 6.2.3. Therefore, we state the result directly without proof.
Theorem 6.2.7. Let G be the join of k-copies of a graph F of order t having P and
P1 as their vertex partitions respectively. If AP1(F) has constant row sum R, then for a
vertex partition P of (G)k(i) such that it is the union of k-copies of vertex partitions P1
of (F)k(i),
EP(G)k(i) = kEP1(F)k(i)− k|R|+ |R+(k−1)t|+(k−1)|R− t|.
The robust P-energy of the k(i)-complement of join of k graphs is given by the fol-
lowing result.
Proposition 6.2.8. Let Gi be graphs of order ni for i = 1,2, . . . ,k. If G = G1▽G2▽ . . .▽
Gk, then for the vertex partition P =V (G)1(i)
EPr(G)1(i) =
k
∑
i=1
ni
∑
j=1
∣∣∣∣λi j +(n−ni)
∣∣∣∣
where (G)1(i) is the 1(i)-complement of the graph G of order n and λi j is an eigenvalue of
APi(G)1(i) for j = 1,2, . . . ,ni, where i = 1,2, . . . ,k.
171
Chapter 6. P-energy of join of graphs
6.3 Some special cases of join of graphs
In this section, we explore the P-energy of join of regular graphs and complete bipartite
graphs. First, we present a lemma that gives a property of AP(G) when G is a regular
graph.
Lemma 6.3.1. If G is a regular graph with vertex partition P = {V1,V2, . . . ,Vk} such
that |Vi|= l, for i = 1,2, . . . ,k, then AP(G) has constant row sum R = l+ 1
n(4m1+2m2−
2m3), where m1 is the number of edges having end vertices in same vertex partition, m2
is the number of edges with end vertices in different partition and m3 is the number of
non-adjacent pairs of vertices in same partition.
Proof. Consider a regular graph G having a vertex partition P = {V1,V2, . . . ,Vk} such
that |Vi| = l for each Vi ∈ P . Since G is a regular graph and each diagonal entry of
AP(G) is l, sum of the elements of each row of AP(G) is a constant, say R. Now we will
determine the value of R in terms of m1, m2 and m3. It can be observed that, number of
2’s, 1’s and −1’s in AP(G) are 2m1, 2m2 and 2m3 respectively. Therefore, the value of R
is given by R = l + 1
n(4m1 +2m2 −2m3).
Remark 6.3.2. If G is an r-regular graph of order n, then for the vertex partitions P1 =
{V (G)} and P2 = {{v1},{v2}, . . . ,{vn}}, the constant row sum of AP1(G) and AP2(G)
is 3r+1 and 2r−n+2 respectively.
Note that, the quantities m1,m2,m3, and m′
1,m
′
2,m
′
3 used in the next theorem are taken
in the same context as that of m1,m2 and m3 mentioned in Lemma 6.3.1.
Theorem 6.3.3. Let G1 and G2 be two regular graphs of order n1 and n2 with vertex
partitions P1 and P2 respectively. If P1 = {U1,U2, . . . ,Uk1} and P2 = {V1,V2, . . . ,Vk2}
172
Chapter 6. P-energy of join of graphs
such that |Ui|= l1 and |Vi|= l2, then for the vertex partition P = P1 ∪P2 of G1▽G2
EP(G1▽G2) = ∑
i=1,2
EPi(Gi)− ∑
i=1,2
∣∣∣∣li + Mi
ni
∣∣∣∣+ 1
2
(|a1|+ |b1|) ,
where M1 = 4m1 + 2m2 − 2m3 and M2 = 4m′
1 + 2m′
2 − 2m′
3, a1 =
[
l1 + l2 +
M1
n1
+ M2
n2
]
+{
(l1 − l2)2 +
(
M1
n1
− M2
n2
)
+2l1
(
M1
n1
− M2
n2
)
+2l2
(
M2
n2
− M1
n1
)
+4n1n2
} 1
2
and b1 =
[
l1 +
l2+
M1
n1
+ M2
n2
]
−
{
(l1− l2)2+
(
M1
n1
− M2
n2
)
+2l1
(
M1
n1
− M2
n2
)
+2l2
(
M2
n2
− M1
n1
)
+4n1n2
} 1
2
.
Proof. For the vertex partition P =P1∪P2 of G1▽G2, by Lemmas 6.1.4 and 6.3.1, the
characteristic polynomial AP(G) is
φP(G1▽G2,λ ) =
φP1(G1,λ )φP2(G2,λ )
{
[λ − (l1 + M1
n1
)][λ − (l2 + M2
n2
)]−n1n2
}
[λ − (l1 + M1
n1
)][λ − (l2 + M2
n2
)]
It can be written as
[λ − (l1 +
M1
n1
)][λ − (l2 +
M2
n2
)]φP(G1▽G2,λ )
=φP1(G1,λ )φP2(G2,λ )
{
[λ − (l1 +
M1
n1
)][λ − (l2 +
M2
n2
)]−n1n2
}
=φP1(G1,λ )φP2(G2,λ )
{
[λ 2 −
[
l1 + l2 +
M1
n1
+
M2
n2
]
λ
+
[
l1 + l2 +
M1
n1
+
M2
n2
]
−n1n2
}
. (6.43)
Let left and right side of Equation (6.43) be L(λ ) and R(λ ) respectively. The roots of the
equations L(λ ) = 0 and R(λ ) = 0 are same. Therefore, the sum of the absolute values of
the roots of these equations are also the same.
173
Chapter 6. P-energy of join of graphs
Thus,
∣∣∣∣(l1 +
M1
n1
)∣∣∣∣+ ∣∣∣∣(l2 +
M2
n2
)∣∣∣∣+EP(G1▽G2,λ ) = EP1(G1,λ )+EP2(G2,λ )+
1
2
{
|a1|+ |b1|
}
,
(6.44)
where M1 = 4m1 + 2m2 − 2m3, M2 = 4m′
1 + 2m′
2 − 2m′
3 and a1, b1 are the roots of the
quadratic equation
{
[λ 2 −
[
l1 + l2 +
M1
n1
+ M2
n2
]
λ +
[
l1 + l2 +
M1
n1
+ M2
n2
]
−n1n2
}
.
Therefore, on simplifying Equation (6.44) we get the required result.
Corollary 6.3.4. Let G1 and G2 be two regular graphs of order n1 and n2, with degrees
r1 and r2 respectively. If P1 and P2 are the vertex partitions of G1 and G2, then corre-
sponding to the vertex partition P = P1 ∪P2 of G1▽G2
(i) for P1 = {V (G1)} and P2 = {V (G2)}
EP(G1▽G2) = ∑
i=1,2
EPi(Gi)− ∑
i=1,2
(3ri +1)+
1
2
{
|a|+ |b|
}
,
where a = [3(r1 + r2)+2]+
√
3(r1 − r2)2 +4n1n2
and b = [3(r1 + r2)+2]−
√
3(r1 − r2)2 +4n1n2,
(ii) for P1 = {{u1},{u2}, . . . ,{un1}} and P2 = {{v1},{v2}, . . . ,{vn2}}
EP(G1▽G2) = ∑
i=1,2
EPi(Gi)− ∑
i=1,2
(2ri −ni +2)+
1
2
{
|a2|+ |b2|
}
,
where a2 = [2(r1+r2)−(n1+n2)+4]+{(2r1+2r2)
2+(n1+n2)
2+4r1(n2−n1)+
4r2(n1−n2)}
1
2 and b2 = [2(r1+ r2)− (n1+n2)+4]−{(2r1+2r2)
2+(n1+n2)
2+
4r1(n2 −n1)+4r2(n1 −n2)}
1
2 .
The subsequent examples are the illustrations of Corollary 6.3.4 in case of the vertex
partition P = P1 ∪P2 of G = G1▽G2 where P1 = {V (G1)} and P2 = {V (G2)}.
174
Chapter 6. P-energy of join of graphs
Example 6.3.5. The complete bipartite graph Kn1,n2 of order n = n1 + n2 is a join of
two null graphs Nn1 and Nn2 of order n1 and n2 respectively. Consider the vertex parti-
tionof Nni is Pi = {V (Nni)}, for i = 1,2 where, V (Nn1) = u1,u2, . . . ,un1 and V (Nn2) =
v1,v2, . . . ,vn2 .
Case 6.3.5.1. If n1 ̸= n2 and r1 = r2 = 0, then by Corollary 6.3.4 (i) and by Theorem
4.3.2, EPr(Nni) = n2
i for i = 1,2 we get
EP(Kn1,n2) = n2
1 +n2
2 +2
√
n1n2 −2. (6.45)
Case 6.3.5.2. If n1 = n2 = r and r1 = r2 = 0, then by Equation (6.45)
EP(Kr,r) =
1
2
[n2 +2n−4], for n = 2r (6.46)
Equation (6.46) can be verified by Theorem 4.3.12.
Example 6.3.6. Let Pi = {V (Kni)} be the vertex partition of Kni for i= 1,2. By Corollary
6.3.4 (i), for the vertex partition P = P1 ∪P2 of Kn1▽Kn2
Case 6.3.6.1. If n1 = n2 = t and r1 = r2 = r, then
EP(Kn1▽Kn2) =2EP1(Kt)−2(3t −2)+ |4t −2|+ |2t −2|= 2EP1(Kt).
Case 6.3.6.2. If n1 ̸= n2 and r1 ̸= r2, then
EP(Kn1▽Kn2) = EP1(Kn1)+EP2(Kn2), for n1,n2 ≥ 2
The next result is required to obtain the P-energy of Kn1▽Cn2 and Cn1▽Cn2 in Exam-
ples 6.3.8 and 6.3.9:
175
Chapter 6. P-energy of join of graphs
Proposition 6.3.7. For a cycle Cn of order n and its vertex partition P = {V (G)},
EPr(Cn) = 7+
n−1
∑
j=1
∣∣∣∣n+1+6cos
(
2π j
n
)∣∣∣∣. (6.47)
Proof. The P-matrix of Cn for P = {V (G)} is a circulant matrix of order n×n with the
first row (n,2,−1, . . . ,−1,2). Therefore, the P-eigenvalues of AP(Cn) are
λ j =

7 if j = 0,
n−1
∏
j=1
∣∣∣∣n+1+6cos
(
2π j
n
)∣∣∣∣ if j = 1,2, . . . ,n−1.
Thus, the result holds.
Example 6.3.8. The P-energy of Kn1▽Cn2 for the vertex partition P = {V1,V2} is
EP(Kn1▽Cn2) = n2
1 +7+
n−1
∑
j=1
∣∣∣∣n2 +1+6cos
(
2π j
n2
)∣∣∣∣,
where V1 is the vertex partition of Kn1 and V2 is the vertex partition of Cn2 .
Example 6.3.9. The P-energy of Cn1▽Cn2 for the vertex partition P = {V1,V2} is
EP(Cn1▽Cn2) =

14+2
n−1
∑
j=1
∣∣∣∣ni +1+6cos
(
2π j
ni
)∣∣∣∣, for n1,n2 < 7
2
√
n1n2 +
n−1
∑
j=1
∣∣∣∣ni +1+6cos
(
2π j
ni
)∣∣∣∣, for n1,n2 ≥ 7
where i = 1,2 and V1 is the vertex partition of Cn1 and V2 is the vertex partition of Cn2 .
By Theorem 6.1.7 and Lemma 6.3.1, we obtain the following theorem, the proof of
which is similar to that of Theorem 6.3.3.
Theorem 6.3.10. Let H be a regular graph of order n1 and P1 = {V1,V2, . . . ,Vt} be its
vertex partition such that |Vi| = l1 for i = 1,2, . . . , t. Let G be the join of k-copies of H
176
Chapter 6. P-energy of join of graphs
and P be its vertex partition such that it is the union of k-copies of P1. Then for P
EP(G) =kEP1(H)+
1
n1
|(k−1)n2
1 + l1 +M1|+
(k−1)
n1
|M1 + l1n1 −n2
1|−
k
n1
|l1n1 +M1|.
Corollary 6.3.11. Let H be an r1-regular graph of order n1 and P1 be its vertex partition.
Let G be the join of k-copies of H. If P is the vertex partition of G such that it is the union
of k-copies of P1, then
(i) EP(G) = kEP1(H)+ |3r1 +1− (k−1)n1|+(k−1)|3r1 +1−n1|− k|3r1 +1|,
when P1 =V (H) and
(ii) EP(G) = kEP1(H)+ |2(r1 +1)− kn1|+2(k−1)|r+1−n1|− k|2r1 −n1 +2|,
when P1 = {{u1},{u2}, . . . ,{un1}}.
Let G be the join of k regular graphs of order n1,n2, . . . ,nk and degree r1,r2, . . . ,rk
respectively. Then the join G is a regular graph if ni − ri = ni+1 − ri+1 [10]. In the next
theorem, we use this condition and obtain the P-energy of a regular graph G which is the
join of k graphs, each of which is regular.
Theorem 6.3.12. Let G1,G2, . . . ,Gk be the regular graphs of order n1,n2, . . . ,nk and de-
gree r1,r2, . . . ,rk respectively. Let P1,P2, . . . ,Pk be their vertex partitions. If G =
G1▽G2▽ . . .▽Gk is an r-regular graph of order n such that r = n− s, then for a vertex
partition P =
k⋃
i=1
Pi of G
EP(G) =
k
∑
i=1
EPi(Gi)−
k
∑
i=1
Ri +
( k−1
∑
i=1
ni +Rk
)
+
k−1
∑
i=1
|Ri −ni| (6.48)
where Ri is a constant row sum of APi(Gi) and s = ni−ri = ni+1−ri+1, for i = 1,2, . . . ,k.
177
Chapter 6. P-energy of join of graphs
Proof. For a vertex partition P = P1 ∪P2 of G, by Lemma 6.1.4,
φP(G1▽G2,λ ) =
φP1(G1,λ )φP2(G2,λ )[λ − (n1 +R2)][λ +n1 −R1]
[λ −R1][λ −R2]
. (6.49)
Now, let G = (G1▽G2)▽G3 and P = P1 ∪P2 ∪P3. Then Equation (6.49) becomes,
φP(G,λ ) =
3
∏
i=i
φPi(Gi,λ )
[λ −Ri]
[λ − (n1 +n2 +R3)][λ +n1 −R1][λ +n2 −R2]
=
3
∏
i=i
φPi(Gi,λ )
[λ −Ri]
[λ − ( ∑
i=1,2
ni +R3)] ∏
i=1,2
[λ +ni −Ri].
Continuing in this way for G1,G2, . . . ,Gk and P =
k⋃
i=1
Pi, we obtain
φP(G,λ ) =
k
∏
i=i
φPi(Gi,λ )
[λ −Ri]
k−1
∏
i=1
[λ − (Ri −ni)]
{
λ −
[ k−1
∑
i=1
ni +Rk
]}
. (6.50)
Now, Equation (6.50) can be written as
k
∏
i=i
[λ −Ri]φP(G,λ ) =
k
∏
i=i
φPi(Gi,λ )
k−1
∏
i=1
[λ − (Ri −ni)]
{
λ −
[ k−1
∑
i=1
ni +Rk
]}
. (6.51)
Consider the left hand side and the right hand side of the Equation (6.51) as S1(λ ) and
S2(λ ) respectively. The roots of equation S1(λ ) = 0 and S2(λ ) = 0 are same. Therefore,
the sum of the absolute values of their roots are also same. Thus,
k
∑
i=i
|Ri|+EP(G,λ ) =
k
∑
i=i
EPi(Gi,λ )+
k−1
∑
i=1
|Ri −ni|+
∣∣∣∣ k−1
∑
i=1
ni +Rk
∣∣∣∣.
Therefore,
EP(G,λ ) =
k
∑
i=i
EPi(Gi,λ )+
k−1
∑
i=1
|Ri −ni|+
∣∣∣∣ k−1
∑
i=1
ni +Rk
∣∣∣∣− k
∑
i=i
|Ri|.
Hence, the result holds.
178
Chapter 6. P-energy of join of graphs
Corollary 6.3.13. Let Gi be a regular graph of order ni, degree ri such that Pi =V (Gi),
for i = 1,2, . . . ,k. If G =
k⋃
i=1
Gi is a regular graph of degree r such that its vertex partition
P =
k⋃
i=1
Pi, then
EP(G) =
k
∑
i=1
EPi(Gi)−
k
∑
i=1
(3ri +1)+
( k−1
∑
i=1
ni +3rk +1
)
+
k−1
∑
i=1
|3ri +1−ni|.
A complete multipartite graph Kt×k is the join of k null graphs of order t. The follow-
ing examples are the particular cases of Corollary 6.3.13.
Example 6.3.14. (i) Let Kt×k = Nt1▽Nt2▽ . . .▽Ntk be a complete multipartite graph of
order n = t × k such that, V (Nti) partition the vertex set of Kt×k and Nti is a null
graph of order ti = t, for i = 1,2, . . .k. Then
EP(Kt×k) = kt2 +2(k−1)t +2(1− k).
(ii) Let Kt×2 be a cocktail party graph of order n which is obtained by taking join of
two null graphs of order t such that n = 2t. Then
EP(Kt×2) = 2t2 +2t −2.
The following remark is a special case of Corollary 6.3.13:
Remark 6.3.15. Let G be an r-regular graph obtained by the join of k-copies of an r1-
regular graph G1 of order n1 such that 3r1 > n1 −1. Then,
EP(G) = kEPr(G1) (6.52)
where EPr(G1) is the robust P-energy of G1 for Pr = {V (G1)}.
179
Chapter 6. P-energy of join of graphs
The following examples illustrate Remark 6.3.15:
Example 6.3.16. (i) The P-energy of join H1 of k-copies of a complete graph Kt is
EP(H1) = kEPr(Kt) = kt2, for t ≥ 2.
(ii) The P-energy of join H2 of k-copies of a complete bipartite graph Kt,t is
EP(H2) = kEPr(Kt,t) = k(2t2 +2t −2), for t ≥ 3,
(iii) The P-energy of join H3 of k-copies of a cycle Ct of order t is
EP(H3) = kEPr(Ct) =

7k+ k
k−1
∑
j=1
∣∣∣∣t +1+6cos
(
2πr
t
)∣∣∣∣, t < 7
2t(k−1)+7(k−2)+ k
k−1
∑
j=1
∣∣∣∣t +1+6cos
(
2πr
t
)∣∣∣∣, t ≥ 7.
Remark 6.3.17. By example 6.3.16 and Equation (6.40) from Theorem 6.2.5 we observe
that,
1. For t ≥ 2, EP(Kt▽Kt▽ . . .▽Kt(k-copies)) = EP(Kt▽Kt▽ . . .▽Kt)k.
2. For t ≥ 3, EP(Kt,t▽Kt,t▽ . . .▽Kt,t(k-copies)) = EP(Kt,t▽Kt,t▽ . . .▽Kt,t)k.
3. For t < 7, EP(Ct▽Ct▽ . . .▽Ct(k-copies)) = EP(Ct▽Ct▽ . . .▽Ct)k.
As a consequence we pose the following problem:
Problem 6.3.18. Characterize the families of graphs satisfying the relation
EP(G1▽G2▽ . . .▽Gk) = EP(G1▽G2▽ . . .▽Gk)k,
where G1,G2, . . . ,Gk may or may not be same.
In the subsequent theorems, we determine the P-energy of join of complete bipartite
graphs using Theorem 6.1.1 and 6.1.7. First we obtain the P-coronal of a complete
bipartite graph.
180
Chapter 6. P-energy of join of graphs
Lemma 6.3.19. Let Kr,s be a complete bipartite graph of order n = r+ s such that r ̸= s.
Then for the vertex partition P = {V1,V2} where V1 and V2 are the two partite sets of Kr,s
and P = {{v1},{v2}, . . . ,{vn}}, the P-coronal of Kr,s is
ΓAP
(λ ) =
nλ +2rs−n
λ 2 −2λ − (rs−1)
. (6.53)
Proof. Let X = diag((λ + s− 1)Ir,(λ + r − 1)Is) be a diagonal matrix of order n× n,
where Ir and Is are identity matrices of order r and s respectively. Then
[λ −AP(Kr,s)]X1n = [λ 2 −2λ− (rs−1)]1n
where 1n is a column vector of order n×1. Therefore,
ΓAP
(λ ) = 1T
n [λ In −AP(Kr,s)]
−11n
=
1T
n X1n
[λ 2 −2λ − (rs−1)]
=
nλ +2rs−n
[λ 2 −2λ − (rs−1)]
.
Theorem 6.3.20. Let Kr1,s1 and K′
r2,s2
be two complete bipartite graphs of order n1 and n2.
Let P1 and P2 be their vertex partitions respectively. Then for a vertex partition P =
P1 ∪P2 of Kr1,s1▽K′
r2,s2
,
EP(Kr1,s1▽K′
r2,s2
) = EP1(Kr1,s1)+EP2(K
′
r2,s2
)−|(1+
√
r1s1)|− |(1−
√
r1s1)|
− |(1+
√
r2s2)|− |(1−
√
r2s2)|+ |c1|+ |c2|+ |c3|+ |c4|
where c1,c2,c3 and c4 are the roots of the quartic polynomial λ 4 − 4λ 3 − [r1s1 + r2s2 +
n1n2 − 6]λ 2 + 2[r1s1(1− n2)+ r2s2(1− n1)+ n1n2 − 2]λ + [r1s1(2n2 − 1)+ r2s2(2n1 −
1)−3r1r2s1s2 −n1n2 +1].
181
Chapter 6. P-energy of join of graphs
Proof. By Lemma 6.3.19, the P-coronal of Kr1,s1 corresponding to AP1(Kr1,s1) is
ΓAP1
(λ ) =
n1λ +2r1s1 −n1
λ 2 −2λ − (r1s1 −1)
. (6.54)
and the P-coronal of K′
r2,s2
corresponding to AP2(K
′
r2,s2
) is
ΓAP2
(λ ) =
n2λ +2r2s2 −n2
λ 2 −2λ − (r2s2 −1)
. (6.55)
By Theorem 6.1.1,
φP(Kr1,s1▽K′
r2,s2
) = φP1(Kr1,s1,λ )φP2(K
′
r2,s2
,λ )[1−ΓAP1
(λ )ΓAP2
(λ )]. (6.56)
Therefore, by Equations (6.54), (6.55) and (6.56)
φP(Kr1,s1▽K′
r2,s2
,λ ) =
φP1(Kr1,s1,λ )φP2(K
′
r2,s2
,λ )
[λ 2 −2λ − (r1s1 −1)][λ 2 −2λ − (r2s2 −1)]
×
{
[λ 2 −2λ − (r1s1 −1)][λ 2 −2λ − (r2s2 −1)]
− [n1λ +2r1s1 −n1][n2λ +2r2s2 −n2]
}
.
It can be written as,
[λ 2 −2λ − (r1s1 −1)][λ 2 −2λ − (r2s2 −1)]φP(Kr1,s1▽K′
r2,s2
,λ )
=φP1(Kr1,s1,λ )φP2(K
′
r2,s2
,λ )
{
[λ 2 −2λ − (r1s1 −1)][λ 2 −2λ − (r2s2 −1)]
− [n1λ +2r1s1 −n1][n2λ +2r2s2 −n2]
}
.
182
Chapter 6. P-energy of join of graphs
Thus,
(λ − (1+
√
r1s1))(λ − (1−
√
r1s1))(λ − (1+
√
r2s2))(λ − (1−
√
r2s2))φP(Kr1,s1▽K′
r2,s2
,λ )
=φP1(Kr1,s1,λ )φP2(K
′
r2,s2
,λ )
{
[λ 2 −2λ − (r1s1 −1)][λ 2 −2λ − (r2s2 −1)]
− [n1λ +2r1s1 −n1][n2λ +2r2s2 −n2]
}
.
On simplifying it further, we obtain
(λ − (1+
√
r1s1))(λ − (1−
√
r1s1))(λ − (1+
√
r2s2))(λ − (1−
√
r2s2))×
φP(Kr1,s1▽K′
r2,s2
,λ ) = φP1(Kr1,s1 ,λ )φP2(K
′
r2,s2
,λ )×
λ
4 −4λ
3 − [r1s1 + r2s2 +n1n2 −6]λ 2 +2[r1s1(1−n2)+ r2s2(1−n1)
+n1n2 −2]λ +[r1s1(2n2 −1)+ r2s2(2n1 −1)−3r1r2s1s2 −n1n2 +1].
Thus,
|(1+
√
r1s1)|+ |(1−
√
r1s1)|+ |(1+
√
r2s2)|+ |(1−
√
r2s2)|+EP(Kr1,s1▽K′
r2,s2
,λ )
= EP1(Kr1,s1,λ )+EP2(K
′
r2,s2
,λ )+ |c1|+ |c2|+ |c3|+ |c4|, (6.57)
where c1,c2,c3 and c4 are the roots of the quartic polynomial
λ
4 −4λ
3 − [r1s1 + r2s2 +n1n2 −6]λ 2 +2[r1s1(1−n2)+ r2s2(1−n1)+n1n2 −2]λ
+[r1s1(2n2 −1)+ r2s2(2n1 −1)−3r1r2s1s2 −n1n2 +1].
Hence, on simplifying Equation (6.57) we get the required result.
Since the proof technique of the following theorem is same as that of Theorem 6.3.20,
we state the Theorem 6.3.21 without proof.
183
Chapter 6. P-energy of join of graphs
Theorem 6.3.21. Let G1 be a graph of order n1 such that AP1(G1) has constant row sum
R1 and Kr1,s1 be a complete bipartite graph of order n2. Let P1 and P2 be their vertex
partitions respectively. Then for a vertex partition P = P1 ∪P2 of G1▽Kr1,s1 ,
EP(G1▽Kr1,s1) =EP1(G1)+EP2(Kr1,s1)−|R1|− |(1+
√
r1s1)|+ |d1|+ |d2|+ |d3|
where d1,d2 and d3 are the roots of the cubic polynomial
λ 3 − (R1 +2)λ 2 − [2R1 − r1s1 −n1n2 +1]λ +[R1(r1s1 −1)−2r1s1s2n1 +n1n2].
In the next theorem, we derive the expression for P-energy of join of k-copies of Kr,s.
Theorem 6.3.22. Let Kr,s be a complete bipartite graph of order t, for r ̸= s and P1 be
its vertex partition. Let G be the join of k-copies of Kr,s and P be its vertex partition such
that it is the union of k-copies of P1. Then
EP(G) =kEP1(Kr,s)−2k
+
1
2
∣∣∣∣t(k−1)+2+
√
t2(k−1)2 +4[2(k−1)+1]rs
∣∣∣∣
+
1
2
∣∣∣∣t(k−1)+2−
√
t2(k−1)2 +4[2(k−1)+1]rs
∣∣∣∣
+(k−1)
√
t2 −4rs.
Proof. By Lemma 6.3.19, the P-coronal of Kr,s corresponding to AP1(Kr,s) is given by
ΓAP1
(λ ) =
tλ +2rs− t
λ 2 −2λ − (rs−1)
. (6.58)
By substituting Equation (6.58) in (6.22), we get
φP(G,λ ) =
[
φP1(Kr,s)
D
]k[
D− (k−1)N
][
D+N
](k−1)
. (6.59)
184
Chapter 6. P-energy of join of graphs
where N = (tλ +2rs− t) and D = λ 2 −2λ − (rs−1). Equation (6.59) can be written as
Dk
φP(G,λ ) =
[
φP1(Kr,s)
]k[
D− (k−1)N
][
D+N
](k−1)
. (6.60)
In order to obtain P-energy, first we need to find the roots of the following terms:
(i) D = λ 2 −2λ − (rs−1),
(ii) [D− (k−1)N] = λ 2 − [t(k−1)+2]λ − [(2(k−1)+1]rs− (k−1)t −1],
(iii) D+N = λ 2 +(t −2)λ +(rs− t +1).
On factorizing these quadratic equations, we get the following quantities. The roots of
(i) D are 1±
√
rs,
(ii)
[
D− (k−1)N
]
are
1
2
{
[t(k−1)+2]±
√
t2(k−1)2 +4[2(k−1)+1]rs
}
,
(iii)
[
D+N
]
are
1
2
[
−[t −2]±
√
t2 +4rs
]
.
Consider the left hand side and the right hand side of the Equation (6.60) as S1(λ ) and
S2(λ ) respectively. The roots of equation S1(λ ) = 0 and S2(λ ) = 0 are same. Therefore,
the sum of the absolute values of their roots are also same. Thus, by Equation (6.60)
k(1+
√
rs)+ k(1−
√
rs)+EP(G) =kEP1(Kr,s)
+
1
2
∣∣∣∣t(k−1)+2+
√
t2(k−1)2 +4[2(k−1)+1]rs
∣∣∣∣
+
1
2
∣∣∣∣t(k−1)+2−
√
t2(k−1)2 +4[2(k−1)+1]rs
∣∣∣∣
+
(k−1)
2
∣∣∣∣−[t −2]+
√
t2 +4rs
∣∣∣∣
+
(k−1)
2
∣∣∣∣−[t −2]−
√
t2 +4rs
∣∣∣∣. (6.61)
185
Chapter 6. P-energy of join of graphs
Since [t −2]<
√
t2 +4rs, on simplifying Equation (6.61), we get
2k+EP(G) =kEP1(Kr,s)
+
1
2
∣∣∣∣t(k−1)+2+
√
t2(k−1)2 +4[2(k−1)+1]rs
∣∣∣∣
+
1
2
∣∣∣∣t(k−1)+2−
√
t2(k−1)2 +4[2(k−1)+1]rs
∣∣∣∣
+
(k−1)
2
{
−[t −2]+
√
t2 +4rs
}
+
(k−1)
2
{
[t −2]+
√
t2 +4rs
}
(6.62)
Therefore, further simplification of Equation (6.62) gives the following equation:
EP(G) =kEP1(Kr,s)−2k
+
1
2
∣∣∣∣t(k−1)+2+
√
t2(k−1)2 +4[2(k−1)+1]rs
∣∣∣∣
+
1
2
∣∣∣∣t(k−1)+2−
√
t2(k−1)2 +4[2(k−1)+1]rs
∣∣∣∣
+(k−1)
√
t2 −4rs.
Hence, the result holds.
In this chapter, we have determined the generalized formula for the characteristic
polynomial of join of two graphs and join of k-copies of a graph, where k is an integer.
Further, we have derived the P-energy of join of two graphs and join of k-copies of a
graph for a special case wherein the P-matrices of the individual graphs have a constant
row sum. We have also dealt with the cases wherein the individual graphs are regular and
complete bipartite graphs. For further study, it would be worth determining the explicit
formula for P-energy of join of any two or more graphs.
186