Resolvent Energy of Unicyclic, Bicyclic and Tricyclic Graphs

Abstract

The resolvent energy of a graph $G$ of order $n$ is defined as $ER={\sum }_{i=1}^n(n-{\lambda }_i{)}^{-1}$, where ${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n$ are the eigenvalues of $G$. In a recent work [Gutman et al., {\it MATCH Commun. Math. Comput. Chem.\/} {\bf 75} (2016) 279--290] the structure of the graphs extremal w.r.t. $ER$ were conjectured, based on an extensive computer--aided search. We now confirm the validity of some of these conjectures.