Remarks on the energy of regular graphs

V. Nikiforov

Published 2016-04-14Version 1

This note is about the energy of regular graphs. The energy of a graph is the sum of the absolute values of the eigenvalues of its adjacency matrix. It is shown that graphs that are close to regular can be made regular with a negligible change of the energy. Also a $k$-regular graph can be extended to a $k$-regular graph of a slightly larger order with almost the same energy. As an application, it is shown that for every sufficiently large $n,$ there exists a regular graph $G$ of order $n$ whose energy $\left\Vert G\right\Vert _{\ast}$ satisfies \[ \left\Vert G\right\Vert _{\ast}>\frac{1}{2}n^{3/2}-n^{13/10}. \] Several infinite families of graphs with maximal or submaximal energy are given, and the energy of almost all regular graphs is determined.