Abstract
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 \(G\), \(\mathcal{E}(G) \geq 2\beta (G)- 2C_{e}(G)\) and so \(\mathcal{E}(G) \geq 2\beta (G)- 2C(G)\), where \(\mathcal{E}(G)\), \(\beta (G)\), \(C_{e}(G)\) 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 \(\mathcal{E}(G), \gamma(G)\) and \(\gamma_t(G)\), where \(\gamma(G)\) and \(\gamma_t(G)\) are domination number and total domination number of \(G\), respectively.
