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arXiv · 2207.04599

A lower bound of the energy of non-singular graphs in terms of average degree

Saieed Akbari·Hossein Dabirian·S. Mahmood Ghasemi
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Abstract

Let $G$ be a graph of order $n$ with adjacency matrix $A(G)$. The \textit{energy} of graph $G$, denoted by $\mathcal{E}(G)$, is defined as the sum of absolute value of eigenvalues of $A(G)$. It was conjectured that if $A(G)$ is non-singular, then $\mathcal{E}(G)\geq\Delta(G)+\delta(G)$. In this paper we propose a stronger conjecture as for $n \geq 5$, $\mathcal{E}(G)\geq n-1+ d$, where $d$ is the average degree of $G$. Here, we show that conjecture holds for bipartite graphs, planar graphs and for the graphs with $d \leq n-2\ln n -3$

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Saieed Akbari, Hossein Dabirian, S. Mahmood Ghasemi. 2022-07-11. A lower bound of the energy of non-singular graphs in terms of average degree. https://arxiv.org/abs/2207.04599

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