Some relations between rank, vertex cover number and energy of graph

چکیده مقاله

 In this paper, we extend some results of [F. Shaveisi, lower
bounds on the vertex cover number and energy of graphs, MATCH
Commun. Math. Comput. Chem, 87(3) (2022) 683-692] which
state some relations between the vertex cover and other parameters,
such as the order and maximum or minimum degree of graphs.
Also, we prove that for a graph G, E(G) ≥ 2β(G) − 2Ce(G) and so
E(G) ≥ 2β(G) − 2C(G), where E(G), β(G), Ce(G) and C(G) denote
the energy, vertex cover, number of even cycles and number of
cycles in G, respectively. For these both inequalities we investigate
their equality. Finally, we give some relations between E(G), γ(G)
and γt(G), where γ(G) and γt(G) are domination number and total
domination number of G, respectively.