On Minimal Energy of Bipartite Unicyclic Graphs of a Given Bipartition

Abstract

The energy of a graph \(G\), denoted by \(E(G)\), is defined to be the sum of absolute values of all eigenvalues of the adjacency matrix of \(G\). Let \(\mathcal{B}(p, q)\) denote the set of bipartite unicyclic graphs with a \((p, q)\)-bipartition, where \(q \geq p \geq 2\). Recently, Li and Zhou [MATCH Commun. Math. Comput. Chem. \(54 (2005) 379-388]\) conjectured that for \(q \geq 3\), \(E(B(3, q)) > E(H(3, q))\), where \(B(3, q)\) and \(H(3, q)\) are respectively graphs as shown in Fig. 1. In this note, we show that this conjecture is true for \(3 \leq q \leq 217\). As a byproduct, we determined the graph with minimal energy among all graphs in \(\mathcal{B}(3, q)\).