Unicyclic graphs with maximal energy

Abstract Let λ 1 , λ 2 , … , λ n be the eigenvalues of a graph G of order n . The energy of G is defined as E ( G ) = | λ 1 | + | λ 2 | + ⋯ + | λ n | . Let P n 6 , 6 be the graph obtained from two copies of C 6 joined by a path P n - 10 , B n be the class of all bipartite bicyclic graphs that are not the graph obtained from two cycles C a and C b ( a , b ⩾ 10 and a ≡ b ≡ 2 (mod 4)) joined by an edge. In this paper, we show that P n 6 , 6 is the graph with maximal energy in B n , which gives a partial solution to Gutman’s conjecture in Gutman and Vidovic (2001) [I. Gutman, D. Vidovic, Quest for molecular graphs with maximal energy: a computer experiment, J. Chem. Inf. Sci. 41 (2001) 1002–1005].

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