On Connected Bipartite $Q$-Integral Graphs

Document Type : Original paper

Authors

Department of Mathematical Sciences, Indian Institute of Technology (BHU) Varanasi, UP, India

Abstract

A graph $G$ is said to be $H$-free if $G$ does not contain $H$ as an induced subgraph. Let $\mathcal{S}_{n}^2(m)$ be a \textit{variation of double star $\mathcal{S}_{n}^2$} obtained by adding m (<=n) disjoint edges between the pendant vertices which are at distance 3 in $\mathcal{S}_{n}^2$. A graph having integer eigenvalues for its signless Laplacian matrix is known as a Q-integral graph. The Q-spectral radius of a graph is the largest eigenvalue of its signless Laplacian. Any connected Q-integral graph G with Q-spectral radius 7 and maximum edge-degree 8 is either $K_{1,4}\square K_2$ or contains $\mathcal{S}_{4}^2(0)$ as an induced subgraph or is a bipartite graph having at least one of the induced subgraphs $\mathcal{S}_{4}^2(m)$, (m=1, 2, 3). In this article, we improve this result by showing that every connected Q-integral graph G having Q-spectral radius 7, maximum edge-degree 8 is always bipartite and $\mathcal{S}_{4}^2(3)$-free.

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References

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Communications in Combinatorics and Optimization

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Available Online from 05 February 2024

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