Abstract
Let D(G) and \({\mathrm{Tr}}(G)\) be, respectively, the distance matrix and the diagonal matrix of the vertex transmissions of a connected graph G. The generalized distance matrix is defined as \(T_{\alpha }(G)=\alpha {\mathrm{Tr}}(G)+(1-\alpha )D(G)\), where \( 0\le \alpha \le 1\). If \(\partial _{1}\ge \partial _{2}\ge \cdots \ge \partial _{n}\) are the eigenvalues of \(T_{\alpha }(G)\), the generalized distance spread (or \(T_{\alpha }\)-spread) is defined as \(S_{T_{\alpha }}(G)=\partial _1-\partial _n\). In this paper, we obtain an upper bound for the smallest generalized distance eigenvalue \(\partial _{n}\) in terms of different graph parameters. In particular, we show that this upper bound is better than the upper bound obtained by Cui et al. (Linear Algebra Appl 563:1–23, 2019). As an application to this upper bound, we obtain a lower bound for the generalized distance spread \(S_{T_{\alpha }}(G)\) and discuss some of its consequences. Furthermore, we obtain a lower bound for \(S_{T_{\alpha }}(G)\) in terms of the chromatic number \(\chi \) of the graph G. Also, we discuss the nature of \(T_{\alpha }\)-spread \(S_{T_{\alpha }}(G)\) under some graph operations.
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Acknowledgements
We are highly grateful to the anonymous referees for their valuable suggestions. The research of S. Pirzada is supported by the SERB-DST research Project number CRG/2020/000109.
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Communicated by Carlos Hoppen.
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Baghipur, M., Ghorbani, M., Ganie, H.A. et al. On the eigenvalues and spread of the generalized distance matrix of a graph. Comp. Appl. Math. 41, 215 (2022). https://doi.org/10.1007/s40314-022-01918-y
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DOI: https://doi.org/10.1007/s40314-022-01918-y
Keywords
- Generalized distance matrix
- Distance signless Laplacian matrix
- Generalized distance spread
- Transmission regular graph