Research ArticleOpen AccessGoogle Scholar indexed
Non-Singular Trees, Unicyclic Graphs and Bicyclic Graphs
Department of Mathematics, Qinghai Nationalities University, Xining, China
Department of Mathematics, Qinghai Nationalities University, Xining, China
Department of Mathematics, Qinghai Nationalities University, Xining, China
- 1 Department of Mathematics, Qinghai Nationalities University, Xining, China
- 2 Department of Mathematics, Qinghai Nationalities University, Xining, China
- 3 Department of Mathematics, Qinghai Nationalities University, Xining, China
Copy link · social · email
Abstract
We called graph G non-singular if adjacency matrix A ( G ) of G is non-singular. A connected graph with n vertices and n -1, n and n +1 edges are called the tree, the unicyclic graph and the bicyclic graph. Respectively, as we all know, each connected bicyclic graph must contain ∞( a , s , b ) or θ ( p , l , q ) as the induced subgraph. In this paper, by using three graph transformations which do not change the singularity of the graph, the non-singular trees, unicyclic graphs and bicyclic graphs are obtained.
KeywordsAdjacency MatrixNon-SingularRankNullity
- Cvetković, D., Doob, M. and Sachs, H. (1980) Spectra of Graphs-Theory and Application. Academic Press, New York.
- Collatz, L. and Sinogowitz, U. (1957) Spektren endlicher Grafen. Abhandlungen aus dem Mathematischen Seminar der Universitat Hamburg, 21, 63-77. https://doi.org/10.1007/BF02941924
- Sciriha, I. and Fowler, P.W. (2008) On Nut and Core Singular Fullerenes. Discrete Mathematics, 308, 267-276. https://doi.org/10.1016/j.disc.2006.11.040
- Brown, M., Kennedy, J.W. and Servatius, B. (1993) Graph Singularity. Graph Theory Notes of New York, 25, 23-32.
- Sciriha, I. (1998) On Singular Line Graphs of Trees. Congressus Numerantium, 135, 73-91.
- Sciriha, I. (2007) A Characterization of Singular Graphs. The Electronic Journal of Linear Algebra, 16, 451-462. https://doi.org/10.13001/1081-3810.1215
- Ashraf, F. and Bamdad, H. (2008) A Note on Graphs with Zero Nullity. MATCH Communications in Mathematical and in Computer Chemistry, 60, 15-19.
- Fan, Y.Z. and Qian, K.S. (2009) On the Nullity of Bipartite Graphs. Linear Algebra and Its Applications, 430, 2943-2949. https://doi.org/10.1016/j.laa.2009.01.007
- Guo, J.M., Yan, W. and Yeh, Y.N. (2009) On the Nullity and the Matching Number of Unicyclic Graphs. Linear Algebra and Its Applications, 431, 1293-1301. https://doi.org/10.1016/j.laa.2009.04.026
- Tang, X. and Liu, B. (2005) On the Nullity of Unicyclic Graphs. Linear Algebra and Its Applications, 408, 212-220. https://doi.org/10.1016/j.laa.2005.06.012
- Hu, S., Tan, X. and Liu, B. (2008) On the Nullity of Bicyclic Graphs. Linear Algebra and Its Applications, 429, 1387-1391. https://doi.org/10.1016/j.laa.2007.12.007
- Chang, A. (2006) On the Trees with Maximum Nullity. MATCH Communications in Mathematical and in Computer Chemistry, 56, 501-508.
- Omidi, G.R. (2009) On the Nullity of Bipartite Graphs. Graphs and Combinatorics, 25, 111-114. https://doi.org/10.1007/s00373-008-0825-5
- Cheng, B. and Liu, B. (2007) On the Nullity of Graphs. The Electronic Journal of Linear Algebra, 16, 60-67. https://doi.org/10.13001/1081-3810.1182
- Marcus, M. and Minc, H. (1964) A Survey of Matrix Theory and Matrix Inequalities. Allyn and Bacon Inc., Boston.
- Ma, H., Yang, W. and Li, S. (2013) Positive and Negative Inertia Index of a Graph. Linear Algebra and Its Applications, 438, 331-341. https://doi.org/10.1016/j.laa.2012.07.014