On Hypoenergetic Unicyclic and Bicyclic Graphs

The energy E(G) of a graph G is the sum of the absolute values of the eigenvalues of G. An n-vertex graph G is said to be hypoenergetic if E(G) < n. In [1], I. Gutman et al. showed that: (a) if Delta = 3, then there exist hypoenergetic trees for n = 4 and n = 7; (b) if Delta = 4, then there exist hypoenergetic trees for all n >= 5 such that n equivalent to k (mod 4), k = 0,1,3; (c) if Delta >= 5, then there exist hypoenergetic trees for all n >= Delta + 1. In this paper we prove that there exist hypoenergetic unicyclic graphs for all n >= 7 and bicyclic graphs for n >= 8. Moreover, we construct hypoenergetic unicyclic and bicyclic graphs for above n.