Abstract
Let \( G \) be an arbitrary simple graph. The energy of \( G \) is defined as the sum of absolute values of all eigenvalues of its adjacency matrix and denoted by \( \mathcal{E}(G) \). Also, the Sombor index of \( G \) is defined as \( SO(G) = \sum\nolimits_{xy\in E(G)} {\sqrt{d_{x}^{2}+d_{y}^{2}}} \), where \( d_x \) and \( d_y \) are the degree of vertices \( x \) and \( y \) in \( G \), respectively. In this paper, we provide the upper and lower bounds for the Sombor index of \( G \) in terms of its energy. For every bipartite graph \( G \), it was proved that \( \mathcal{E}(G) \leq \sqrt{2/\delta^3(G)}SO(G) \), where \( \delta \) is a minimum degree of \( G \). We show that this result holds for any arbitrary graph. Also, we prove \( \mathcal{E}(G)\leq SO(G)/ (\sqrt{2}\delta(G)) \), if \( \delta (G)\geq 4 \). Moreover, we show that \( \sqrt{\mathcal{E}(G)} \geq SO(G)/\sqrt{m\Delta^3(G)} \), where \( \Delta \) and \( m \) are maximum degree and size of \( G \), respectively. Furthermore, we improve some of the stated inequalities between energy and degree based indices of graphs, like the first Zagreb index and the forgotten index, in the existing literature.
