Energy of graphs containing disjoint cycles

Let G be a graph. The energy E(G) is the sum of the absolute values of the eigenvalues of the adjacency matrix of G. In [Energy, matching number and odd cycles of graphs, Linear Algebra Appl. 577 (2019) 159{167] it has been proved that for a graph G whose cycles are odd and vertex disjoint, if from each cycle of G, we remove an arbitrary edge to obtain a tree T, then E(G) ≥ E(T). There is a gap in the proof. In this paper, we correct the proof and generalize this result by showing that if G is a graph all of whose cycles are vertex disjoint and the length of each cycle is not 0, modulo 4, then for any spanning tree of G, E(G) ≥ E(T). Finally we give an upper bound on E(G) of a graph G all of whose cycles are vertex disjoint.