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Complete Solution to a Problem on the Maximal Energy of Unicyclic Bipartite Graphs
| Content Provider | Semantic Scholar |
|---|---|
| Author | Huo, Bofeng Li, Xueliang Shi, Yongtang |
| Copyright Year | 2010 |
| 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 Cn the cycle, and P 6 n the unicyclic graph obtained by connecting a vertex of C6 with a leaf of Pn−6 . Caporossi et al. conjecture that the unicyclic graph with maximal energy is P 6 n for n = 8, 12, 14 and n ≥ 16. In“Y. Hou, I. Gutman and C. Woo, Unicyclic graphs with maximal energy, Linear Algebra Appl. 356(2002), 27–36”, the authors proved that E(P 6 n ) is maximal within the class of the unicyclic bipartite n-vertex graphs differing from Cn . And they also claimed that the energy of Cn and P 6 n 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 6 n is greater than that of Cn for n = 8, 12, 14 and n ≥ 16, which completely solves this open problem and partially solves the above conjecture. |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | https://ia601002.us.archive.org/35/items/arxiv-1010.6129/1010.6129.pdf |
| Language | English |
| Access Restriction | Open |
| Subject Keyword | APPL1 gene Adjacency matrix Carrier-to-noise ratio Graph (discrete mathematics) Graph - visual representation Linear algebra Maximal set Pseudoforest Vertex |
| Content Type | Text |
| Resource Type | Article |
