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 $q\ge p\ge 2$. Recently, Li and Zhou [MATCH Commun. Math. Comput. Chem. $54(2005)379-388]$ conjectured that for $q\ge 3$, $E(B(3,q))>E(H(3,q))$, 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 $3\le q\le 217$. As a byproduct, we determined the graph with minimal energy among all graphs in $\mathcal{B}(3,q)$.