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On the nullity of a family of tripartite graphs. (English) Zbl 1339.05229
Summary: The eigenvalues of the adjacency matrix of a graph form the spectrum of the graph. The multiplicity of the eigenvalue zero in the spectrum of a graph is called nullity of the graph. Y.-Z. Fan and K.-S. Qian [Linear Algebra Appl. 430, No. 11–12, 2943–2949 (2009; Zbl 1169.05346)] obtained the nullity set of \(n\)-vertex bipartite graphs and characterized the bipartite graphs with nullity \(n-4\) and the regular \(n\)-vertex bipartite graphs with nullity \(n-6\). In this paper, we study similar problem for a class of tripartite graphs. As observed the nullity problem in tripartite graphs does not follow as an extension to that of the nullity of bipartite graphs, this makes the study of nullity in tripartite graphs interesting. In this direction, we obtain the nullity set of a class of \(n\)-vertex tripartite graphs and characterize these tripartite graphs with nullity \(n-4\). We also characterize some tripartite graphs with nullity \(n-6\) in this class.
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
05C38 Paths and cycles
Full Text: DOI
[1] B. Cheng, B. Liu, On the nullity of graphs, El. J. Lin. Algebra16 (2007) 60-67. ⇒99 · Zbl 1142.05336
[2] L. Collatz, U. Sinogowitz, Spektren endlicher grafen, Abh. Math. Sere. Univ. Hamburg21 (1957), 63-77. ⇒97 · Zbl 0077.36704
[3] Y.-Z. Fan, Yue Wang, Yi Wang, A note on the nullity of unicyclic signed graphs, Linear Algebra Appl.438 (2013) 1193-1200. ⇒97 · Zbl 1257.05083
[4] Y.-Z. Fan, K.-S. Qian, On the nullity of bipartite graphs, Linear Algebra Appl.430 (2009) 2943-2949. ⇒97, 99 · Zbl 1169.05346
[5] S. Fiorini, I. Gutman, I. Sciriha, Trees with maximum nullity, Linear Algebra Appl.397 (2005) 245-251. ⇒97 · Zbl 1068.05015
[6] S. Hu, T. Xuezhong, B. Liu, On the nullity of bicyclic graphs, Linear Algebra Appl.429 (2008) 1387-1391. ⇒97 · Zbl 1144.05319
[7] W. Li, A. Chang, On the trees with maximum nullity, MATCH Commun. Math. Comput. Chem.56, 3 (2006) 501-508. ⇒97 · Zbl 1121.05072
[8] W. Li, A. Chang, Describing the nonsingular unicyclic graph, J. Math. Study4 (2007) 442-445. ⇒97 · Zbl 1150.05037
[9] J. Li, A. Chang, W.C. Shiu, On the nullity of bicyclic graphs, MATCH Commun. Math. Comput. Chem.60, 1 (2008) 21-36. ⇒97 · Zbl 1199.05229
[10] S. Pirzada, An Introduction to Graph Theory, Universities Press, Orient Black-Swan, Hyderabad, India, 2005. ⇒97
[11] T. Xuezhong, B. Liu, On the nullity of unicyclic graphs, Linear Algebra Appl.408 (2005), 212-220. ⇒97 · Zbl 1073.05044
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