Let G be a graph of order n. For i=1,2,…,n, let di be the degree of the vertex vi of G. The extended adjacency matrix Aex of G is defined so that its (i, j)-entry is equal to 12(didj+djdi) if the ...vertices vi and vj are adjacent, and 0 otherwise,Yang et al. (1994). The spectral radius η1 and the energy Eex of the Aex-matrix are examined. Lower and upper bounds on η1 and Eex are obtained, and the respective extremal graphs characterized.
Let G be a bipartite graph of order n with m edges. The energy E(G) of G is the sum of the absolute values of the eigenvalues of the adjacency matrix A. In 1974, one of the present authors ...established lower and upper bounds for E(G) in terms of n, m, and detA. Now, more than 40 years later, we correct some details of this result and determine the extremal graphs. In addition, an upper bound on the Laplacian energy of bipartite graphs in terms of n, m, and the first Zagreb index is obtained, and the extremal graphs characterized.
Spectra of Deza graphs Akbari, S.; Ghodrati, A. H.; Hosseinzadeh, M. A. ...
Linear & multilinear algebra,
01/2022, Volume:
70, Issue:
2
Journal Article
Peer reviewed
A Deza graph with parameters
is a k-regular graph with n vertices such that any two of its vertices have b or a common neighbours, where
. In this paper we investigate spectra of Deza graphs. In ...particular, using the eigenvalues of a Deza graph we determine the eigenvalues of its children. Divisible design graphs are significant cases of Deza graphs. Sufficient conditions for Deza graphs to be divisible design graphs are given, a few families of divisible design graphs are presented and their properties are studied. Our special attention goes to the invertibility of the adjacency matrices of Deza graphs.
On energy of line graphs Das, Kinkar Ch; Mojallal, Seyed Ahmad; Gutman, Ivan
Linear algebra and its applications,
06/2016, Volume:
499
Journal Article
Peer reviewed
Open access
The energy E(G) of a simple graph G is the sum of absolute values of the eigenvalues of its adjacency matrix. The line graph of G is denoted by LG. We obtain a relation between E(LG) and E(G), and ...establish bounds on E(LG). Moreover, we present results pertaining to equienergetic and hyperenergetic graphs.
Chromatic spectrum of a colored graph G is a multiset of eigenvalues of colored adjacency matrix of G. The nullity of a disconnected graph is equal to sum of nullities of its components but we show ...that this result does not hold for colored graphs. In this paper, we investigate the chromatic spectrum of three different classes of 2-regular bipartite colored graphs. In these classes of graphs, it is proved that the nullity of G is not sum of nullities of components of G. We also highlight some important properties and conjectures to extend this problem to general graphs.
Spectra of network related graphs have numerous applications in computer sciences, electrical networks and complex networks to explore structural characterization like stability and strength of these ...different real-world networks. In present article, our consideration is to compute spectrum based results of generalized prism graph which is well-known planar and polyhedral graph family belongs to the generalized Petersen graphs. Then obtained results are applied to compute some network related quantities like global mean-first passage time, average path length, number of spanning trees, graph energies and spectral radius.
Let
G
be an
n
-vertex graph. If
λ
1
,
λ
2
,
…
,
λ
n
and
μ
1
,
μ
2
,
…
,
μ
n
are the ordinary (adjacency) eigenvalues and the Laplacian eigenvalues of
G
, respectively, then the Estrada index and the ...Laplacian Estrada index of
G
are defined as
EE
(
G
)
=
∑
i
=
1
n
e
λ
i
and
LEE
(
G
)
=
∑
i
=
1
n
e
μ
i
, respectively. Some new lower bounds for
EE
and
LEE
are obtained and shown to be the best possible.
On the Laplacian-energy-like invariant Das, Kinkar Ch; Gutman, Ivan; Çevik, A. Sinan
Linear algebra and its applications,
02/2014, Volume:
442
Journal Article
Peer reviewed
Open access
Let G be a connected graph of order n with Laplacian eigenvalues μ1⩾μ2⩾⋯⩾μn-1>μn=0. The Laplacian-energy-like invariant of the graph G is defined as
LEL=LEL(G)=∑i=1n-1μi.
Lower and upper bounds for ...LEL are obtained, in terms of n, number of edges, maximum vertex degree, and number of spanning trees.
Let G be a connected graph of order n and size m with Laplacian eigenvalues μ1≥μ2≥⋯≥μn=0. The Kirchhoff index of G, denoted by Kf, is defined as: Kf=n∑i=1n−11μi. The Laplacian-energy-like invariant ...(LEL) and the Laplacian energy (LE) of the graph G, are defined as: LEL=∑i=1n−1μi and LE=∑i=1n|μi−2mn|, respectively. We obtain two relations on LEL with Kf, and LE with Kf. For two classes of graphs, we prove that LEL>Kf. Finally, we present an upper bound on the ratio LE/LEL and characterize the extremal graphs.
PDF
