On the Energy and the First Zagreb Index of Graphs

Let [Formula: see text] be a graph of order [Formula: see text] and size [Formula: see text]. In this paper, we derive both lower and upper bounds for the energy of a graph in terms of the average of the [Formula: see text] singular values of [Formula: see text]. Using these results, we obtain lower and upper bounds for the graph energy in terms of the first Zagreb index [Formula: see text], the clique number [Formula: see text], and the parameter [Formula: see text], defined as the largest integer such that the star graph [Formula: see text] is an induced subgraph of [Formula: see text]. Recently, Das et al. conjectured that the graph energy [Formula: see text] satisfies [Formula: see text]. We also present families of graphs that satisfy this conjecture. In particular, we show that [Formula: see text] for all non-singular graphs having no eigenvalues in the interval [Formula: see text].