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 which invariants or graph properties can be deduced from 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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Acknowledgements
The research of Santanu Mandal is supported by the University Grants Commission of India under the beneficiary code BININ01569755. This author also acknowledges the infrastructure provided by the VIT Bhopal University. The research of Zoran Stanić is supported by the Fund of the Republic of Serbia; grant number 7749676: Spectrally Constrained Signed Graphs with Applications in Coding Theory and Control Theory – SCSG-ctct.
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Mandal, S., Mehatari, R. & Stanić, Z. Laplacian eigenvalues and eigenspaces of cographs generated by finite sequence. Indian J Pure Appl Math (2024). https://doi.org/10.1007/s13226-024-00572-w
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DOI: https://doi.org/10.1007/s13226-024-00572-w