Abstract
The Sombor index of graph \( G \) is defined as \( \sum\nolimits_{uv\in E(G)} \sqrt{d_{u}^{2}+d_{v}^{2}} \), where \( d_{u}\) and \(d_{v}\) are the degree of vertices \(u\) and \(v\) in \( G \), respectively. The energy of \( G \) is defined as the sum of absolute values of all eigenvalues of its adjacency matrix and denoted by \( \mathcal{E}(G) \). It was proved that if \( G \) is a graph of order at least \( 3 \), then \( \mathcal{E}(G) < SO(G) \). In this paper, we strengthen this result by showing that if \( G \) is a connected graph of order \( n \) which is not \( P_{n} (n\leq 8)\), then \( \mathcal{E}(G) \leq \dfrac{SO(G) }{2}\).
