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arXiv · 2305.04252

Laplacian eigenvalues and eigenspaces of cographs generated by finite sequence

Santanu Mandal·Ranjit Mehatari·Zoran Stanic
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Abstract

In this paper we consider particular graphs defined by $\overline{\overline{\overline{K_{\alpha_1}}\cup K_{\alpha_2}}\cup\cdots \cup K_{\alpha_k}}$, where $k$ is even, $K_\alpha$ is a complete graph on $\alpha$ vertices, $\cup$ stands for the disjoint union and an overline denotes the complementary graph. These graphs do not contain the $4$-vertex path as an induced subgraph, i.e., they belong to the class of cographs. In addition, they are iteratively constructed from the generating sequence $(\alpha_1, \alpha_2, \ldots, \alpha_k)$. Our primary question is what invariants or graph properties can be deduced form a given sequence. In this context, we compute the Lapacian eigenvalues and the corresponding eigenspaces, and derive a lower and an upper bound for the number of distinct Laplacian eigenvalues. We also determine the graphs under consideration with a fixed number of vertices that either minimize or maximize the algebraic connectivity (that is the second smallest Laplacian eigenvalue). The clique number is computed in terms of a generating sequence and a relationship between it and the algebraic connectivity is established.

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Santanu Mandal, Ranjit Mehatari, Zoran Stanic. 2023-05-07. Laplacian eigenvalues and eigenspaces of cographs generated by finite sequence. https://arxiv.org/abs/2305.04252

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