More on a Conjecture about Tricyclic Graphs with Maximal Energy *

Abstract The energy E(G) of a simple graph G is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix. This concept was introduced by I. Gutman in 1977. Recently, Aouchiche et al. proposed a conjecture about tricyclic graphs: If G is a tricyclic graphs on n vertices with n = 20 or n ≥ 22, then E(G) ≤ E(P 6,6,6 n ) with equality if and only if G ∼ = P 6,6,6 n , where P 6,6,6 n denotes the graph with n ≥ 20 vertices obtained from three copies of C 6 and a path P n−18 by adding a single edge between each of two copies of C 6 to one endpoint of the path and a single edge from the third C 6 to the other endpoint of the P n−18 . Li et al. [X. Li, Y. Shi, M. Wei, J. Li, On a conjecture about tricyclic graphs with maximal energy, MATCH Commun. Math. Comput. Chem. 72 (2014) 183-214] proved that the conjecture is true for graphs in the graph class G(n; a, b, k), where G(n; a, b, k) denotes the set of all connected bipartite tricyclic graphs on n ≥ 20 vertices with three vertex-disjoint cycles C a , C b and C k , apart from 9 subclasses of such graphs. In this paper, we improve the above result and prove that apart from 7 smaller subclasses of such graphs the conjecture is true for graphs in the graph class G(n; a, b, k)