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Proof of conjectures on adjacency eigenvalues of graphs
- Source :
-
Discrete Mathematics . Jan2013, Vol. 313 Issue 1, p19-25. 7p. - Publication Year :
- 2013
-
Abstract
- Abstract: Let be a simple graph of order with triangle(s). Also let be the eigenvalues of the adjacency matrix of graph . X. Yong [X. Yong, On the distribution of eigenvalues of a simple undirected graph, Linear Algebra Appl. 295 (1999) 73–80] conjectured that (i) is complete if and only if and also (ii) is complete if and only if . Here we disprove this conjecture by a counter example. Wang et al. [J.F. Wang, F. Belardo, Q.X. Huang, B. Borovićanin, On the two largest Q-eigenvalues of graphs, Discrete Math. 310 (2010) 2858–2866] conjectured that friendship graph is determined by its adjacency spectrum. Here we prove this conjecture. The eccentricity of a vertex is the maximum distance from it to another vertex and the average eccentricity of a graph is the mean value of eccentricities of all vertices of . Moreover, we mention three conjectures, obtained by the system AutoGraphiX, about the average eccentricity , girth and the spectral radius of graphs (see Aouchiche (2006) [1], available online at http://www.gerad.ca/~agx/). We give a proof of one conjecture and disprove two conjectures by counter examples. [Copyright &y& Elsevier]
Details
- Language :
- English
- ISSN :
- 0012365X
- Volume :
- 313
- Issue :
- 1
- Database :
- Academic Search Index
- Journal :
- Discrete Mathematics
- Publication Type :
- Academic Journal
- Accession number :
- 82599024
- Full Text :
- https://doi.org/10.1016/j.disc.2012.09.017