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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.
MSC:
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
05C38 Paths and cycles
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