Abstract
Let \(G\) be a graph with \(n\) vertices and \(m\) edges. A vertex cover of \(G\) is a set \(Q\) of vertices that contains one endpoint of every edge; the minimum cardinality of a vertex cover is called the vertex cover number and it is denoted by \(\beta\). In this paper, we prove new lower bounds for the vertex cover number. For example it is shown that \(\beta \geq \frac{n}{\Delta + 1}\) and \(\beta \geq \frac{\sqrt{(2\Delta - 1)^2 + 8m} - (2\Delta - 1)}{2}\), where \(\Delta\) denotes the maximum degree of \(G\). Moreover, both of these inequalities turn into equalities if and only if \(G\) is a star graph. Some new lower bounds for the energy of \(G\) are given, too.