A proof of the conjecture on hypoenergetic graphs with maximum degree $\Delta \leq 3$

TL;DR

This paper confirms the conjecture that among connected graphs with maximum degree at most 3 containing a quadrangle, only the complete bipartite graph K_{2,3} is hypoenergetic, based on spectral graph theory analysis.

Contribution

It provides a proof confirming the uniqueness of K_{2,3} as the hypoenergetic graph under the specified conditions.

Findings

K_{2,3} is the only hypoenergetic connected quadrangle-containing graph with Δ ≤ 3

The conjecture by Majstorović et al. is validated

Spectral properties characterize the hypoenergetic graphs in this class

Abstract

The energy $E(G)$ of a graph $G$ is defined as the sum of the absolute values of its eigenvalues. A graph $G$ of order $n$ is said to be hypoenergetic if $E(G)<n$. Majstorovi\'{c} et al. conjectured that complete bipartite graph $K_{2,3}$ is the only hypoenergetic connected quadrangle-containing graph with maximum degree $\Delta \leq 3$. This paper is devoted to giving a confirmative proof to the conjecture.