Affiliation:
1. Department of Mathematical Sciences, College of Science, United Arab Emirate University, Al Ain 15551 , Abu Dhabi , UAE
Abstract
Abstract
For a simple graph with vertex set
{
v
1
,
v
2
,
…
,
v
n
}
\left\{{v}_{1},{v}_{2},\ldots ,{v}_{n}\right\}
and degree sequence
d
v
i
i
=
1
,
2
,
…
,
n
{d}_{{v}_{i}}\hspace{0.33em}i=1,2,\ldots ,n
, the inverse sum indeg matrix (ISI matrix)
A
ISI
(
G
)
=
(
a
i
j
)
{A}_{{\rm{ISI}}}\left(G)=\left({a}_{ij})
of
G
G
is a square matrix of order
n
,
n,
where
a
i
j
=
d
v
i
d
v
j
d
v
i
+
d
v
j
,
{a}_{ij}=\frac{{d}_{{v}_{i}}{d}_{{v}_{j}}}{{d}_{{v}_{i}}+{d}_{{v}_{j}}},
if
v
i
{v}_{i}
is adjacent to
v
j
{v}_{j}
and 0, otherwise. The multiset of eigenvalues
τ
1
≥
τ
2
≥
⋯
≥
τ
n
{\tau }_{1}\ge {\tau }_{2}\hspace{0.33em}\ge \cdots \ge {\tau }_{n}
of
A
ISI
(
G
)
{A}_{{\rm{ISI}}}\left(G)
is known as the ISI spectrum of
G
G
. The ISI energy of
G
G
is the sum
∑
i
=
1
n
∣
τ
i
∣
\mathop{\sum }\limits_{i=1}^{n}| {\tau }_{i}|
of the absolute ISI eigenvalues of
G
.
G.
In this article, we give some properties of the ISI eigenvalues of graphs. Also, we obtain the bounds of the ISI eigenvalues and characterize the extremal graphs. Furthermore, we construct pairs of ISI equienergetic graphs for each
n
≥
9
n\ge 9
.
Subject
Geometry and Topology,Algebra and Number Theory
Reference28 articles.
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3. S. Pirzada, H. A. Ganie, B. A. Rather, and R. Ul Shaban, On α-adjacency energy of graphs and Zagreb index, AKCE Int. J. Graphs Comb. 18 (2021), no. 1, 39–46.
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Cited by
2 articles.
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