Affiliation:
1. Department of Mathematics, Kuwait University , Safat 13060 , Kuwait
2. Faculty of Information Studies , 8000 Novo Mesto , Slovenia
Abstract
Abstract
The Collatz-Sinogowitz irregularity index is the oldest known numerical measure of graph irregularity. For a simple and connected graph
G
G
of order
n
n
and size
m
m
, it is defined as
CS
(
G
)
=
λ
1
−
2
m
/
n
,
\hspace{0.1em}\text{CS}\hspace{0.1em}\left(G)={\lambda }_{1}-2m\hspace{0.1em}\text{/}\hspace{0.1em}n,
where
λ
1
{\lambda }_{1}
is the largest eigenvalue of the adjacency matrix of
G
G
, and
2
m
/
n
2m\hspace{0.1em}\text{/}\hspace{0.1em}n
is the average vertex degree of
G
G
. Here, the Collatz-Sinogowitz inverse irregularity problem is studied. For every integer
i
≥
0
i\ge 0
, it is shown that there exists a graph
G
G
such that
CS
(
G
)
=
i
\hspace{0.1em}\text{CS}\hspace{0.1em}\left(G)=i
. Also, for every interval
I
i
=
(
i
,
i
+
1
)
{I}_{i}=\left(i,i+1)
, it is shown that there are infinitely many graphs whose Collatz-Sinogowitz irregularity lies in
I
i
{I}_{i}
.
Reference11 articles.
1. L. Collatz and U. Sinogowitz, Spektren endlicher Graphen, Abh. Math. Semin. Univ. Hambg. 21 (1957), 63–77.
2. F. K. Bell, A note on the irregularity of graphs, Linear Algebra Appl. 161 (1992), 45–54.
3. M. O. Albertson, The irregularity of a graph, Ars Combin. 46 (1997), 219–225.
4. I. Gutman, M. Togan, A. Yurttas, A. S. Cevik, and I. N. Cangul, Inverse problem for sigma index, MATCH Commun. Math. Comput. Chem. 79 (2018), 491–508.
5. D. Dimitrov and D. Stevanović, On the σt-irregularity and the inverse irregularity problem, Appl. Math. Comput. 441 (2023), 127709.