Complete solution to a problem on the maximal energy of unicyclic bipartite graphs
By Bofeng Huo, Xueliang Li, Yongtang Shi. 2010
53 citations recorded by the source corpus.
Abstract The energy of a simple graph G , denoted by E ( G ) , is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix. Denote by C n the cycle, and P n 6 the unicyclic graph obtained by connecting a vertex of C 6 with a leaf of P n - 6 . Caporossi et al. conjectured that the unicyclic graph with maximal energy is P n 6 for n = 8 , 12 , 14 and n ≥ 16 . In Hou et al. (2002) [Y. Hou, I. Gutman, C. Woo, Unicyclic graphs with maximal energy, Linear Algebra Appl. 356 (2002) 27–36], the authors proved that E ( P n 6 ) is maximal within the class of the unicyclic bipartite n -vertex graphs differing from C n . And they also claimed that the energies of C n and P n 6 is quasi-order incomparable and left this as an open problem. In this paper, by utilizing the Coulson integral formula and some knowledge of real analysis, especially by employing certain combinatorial techniques, we show that the energy of P n 6 is greater than that of C n for n = 8 , 12 , 14 and n ≥ 16 , which completely solves this open problem and partially solves the above conjecture.
Subfields: Geometry and Topology, Materials Chemistry, Organic Chemistry
View Paper: https://www.semanticscholar.org/paper/e79e445a06fe8cd5bedef2a37d1dbc1f4191166d
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Citations in the corpus
- On a Conjecture about Tricyclic Graphs with Maximal Energy (2013, background): “[20] and Andriantiana [1] independently proved that E(Cn) < E(P 6 n), and then completely determined that P 6 n is the only graph which attains the maximum value of the energy among all the unicyclic bipartite graphs for n = 8, 12, 14 and n ≥ 16, which partially solves the above conjecture.”