The energy of a graph G, denoted by E(G), is defined as the sum of the absolute values of all eigenvalues of G. In this paper we present some new lower bounds for energy of non-singular graphs, ...connected non-singular graphs and connected unicyclic non-singular graphs in terms of number of vertices, number of edges, maximum degree and Zagreb indices.
Let
G
be a graph of order
n
, and let
Δ
(
G
)
,
δ
(
G
)
and
d
¯
be the maximum, minimum and average degrees of
G
, respectively. In 2020, Akbari and Hosseinzadeh proposed a conjecture that
E
(
G
)
≥
...Δ
(
G
)
+
δ
(
G
)
for all non-singular graphs
G
. Recently, they gave a strengthened version claiming that
E
(
G
)
≥
n
-
1
+
d
¯
for all non-singular graphs
G
, except two counterexamples of order 4. They proved this new conjecture for regular graphs, bipartite graphs, planar graphs and graphs with some other special properties. In this paper, we continue the study of the conjecture and find that it is true for the family of threshold graphs of order at least 5.
A graph is said to be NSSD (=non-singular with a singular deck) if it has no eigenvalue equal to zero, whereas all its vertex-deleted subgraphs have eigenvalues equal to zero. NSSD graphs are of ...importance in the theory of conductance of organic compounds. In this paper, a novel method is described for constructing NSSD molecular graphs from the commuting graphs of the
-group. An algorithm is presented to construct the NSSD graphs from these commuting graphs.
Lower bounds for the energy of graphs Jahanbani, Akbar
AKCE International Journal of Graphs and Combinatorics,
April 2018, 4/1/2018, 2018-04-01, Letnik:
15, Številka:
1
Journal Article
Recenzirano
Odprti dostop
Let G be a finite simple undirected graph with n vertices and m edges. The energy of a graph G, denoted by E(G), is defined as the sum of the absolute values of the eigenvalues of G. In this paper we ...present lower bounds for E(G) in terms of number of vertices, edges, Randić index, minimum degree, diameter, walk and determinant of the adjacency matrix. Also we show our lower bound in (11) under certain conditions is better than the classical bounds given in Caporossi et al. (1999), Das (2013) and McClelland (1971).
A graph is said to be non-singular if it has no eigenvalue equal to zero; otherwise it is singular. Molecular graphs that are non-singular and also have the property that all subgraphs of them ...obtained by deleting a single vertex are themselves singular, known as NSSD graphs, are of importance in the theory of molecular π-electron conductors; NSSD = non-singular graph with a singular deck. Whereas all non-singular bipartite graphs (therefore, the molecular graphs of all closed-shell alternant conjugated hydrocarbons) are NSSD, the non-bipartite case is much more complicated. Only a limited number of non-bipartite molecular graphs have the NSSD property. Several methods for constructing such molecular graphs are presented.
PDF
