Affiliation:
1. Departamento de Engenharia de Produção - Centro Federal de Educação Tecnológica do Rio de Janeiro , Rio de Janeiro , 20271-110 , Brazil
2. Departamento de Matemática - Colégio Pedro II , Rio de Janeiro , 20930-380 , Brazil
3. Departamento de Matemática - Escola Nacional de Ciências Estatiiisticas , Rio de Janeiro , 20231-050 , Brazil
Abstract
Abstract
Let
G
G
be a simple graph with adjacency matrix
A
(
G
)
A\left(G)
, degree diagonal matrix
D
(
G
)
,
D\left(G),
and let
l
(
G
)
l\left(G)
be the line graph of
G
G
. In 2017, Nikiforov defined the
A
α
{A}_{\alpha }
-matrix of
G
G
,
A
α
(
G
)
{A}_{\alpha }\left(G)
, as a linear convex combination of
A
(
G
)
A\left(G)
and
D
(
G
)
D\left(G)
, in the following way,
A
α
(
G
)
≔
α
A
(
G
)
+
(
1
−
α
)
D
(
G
)
,
{A}_{\alpha }\left(G):= \alpha A\left(G)+\left(1-\alpha )D\left(G),
where
α
∈
[
0
,
1
]
\alpha \in \left[0,1]
. In this study, we present some bounds for the largest eigenvalue of
A
α
(
G
)
,
{A}_{\alpha }\left(G),
and for some eigenvalues of
A
α
(
l
(
G
)
)
{A}_{\alpha }\left(l\left(G))
. Extremal graphs attaining some of these bounds are also characterized. Furthermore, some comparisons between the new bounds obtained in this study, and between these and some bounds presented by Nikiforov are made.