On spectral radius of the generalized distance matrix of a graph. (English) Zbl 1513.05139

Summary: If \(Tr(G)\) and \(D(G)\) are respectively the diagonal matrix of vertex transmission degrees and distance matrix of a connected graph \(G\), the generalized distance matrix \(D_{\alpha}(G)\) is defined as \(D _{\alpha}(G) = \alpha T r(G) + (1- \alpha) D(G)\), where \(0 \leq \alpha \leq 1\). We obtain an upper bound for the spectral radius \(\partial(G)\) (largest eigenvalue) of \(D_{\alpha}(G)\) as \[ \partial (G) \leq \underset {1\le i, j \le n} \max \, \frac{1}{2} \left[\alpha t_i + t_j - (1-\alpha)d_{ij} + \sqrt{(\alpha t_i - t_j)^2 + (1-\alpha)(1-\alpha - 2t_j -4t_i - 2 \alpha t_i)d_{ij}}\right] \] where \(t_{\max}=t_1 \geq t_2 \geq \cdots \geq t_n= t_{\min}\) are the vertex transmission degrees of \(G\) and \(d_{ij}\) is the distance between the vertices \(v_i\), \(v_j \in G\). Further, we show the existence of graphs for which equality holds.


05C12 Distance in graphs
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
15A18 Eigenvalues, singular values, and eigenvectors
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[1] H. Ahmad, A. Alhevaz, M. Baghipur and Gui-Xian Tian, Bounds for generalized distance spectral radius and the entries of the principal eigenvector, Tamkang J. Math.52(2021) 69-89. · Zbl 1476.05105
[2] A. Brouwer, W. Haemers,Spectra of Graphs, Springer, New York, 2012. · Zbl 1231.05001
[3] S. Y. Cui, J. X. He, G. X. Tian, The generalized distance matrix,Linear Algebra Appl.563(2019) 1-23. · Zbl 1403.05083
[4] L. DeVille, The generalized distance spectrum of a graph and applications,Linear Multilin. Algebra, DOI: 10.1080/03081087.2020.1803187, In press. · Zbl 1497.05145
[5] R. C. Diaz, G. Pasten and O. Rojo, New results on theDα-matrix of connected graphs,Linear Algebra Appl.577(2019) 168-185. · Zbl 1416.05170
[6] H. Guo, B. Zhou, On the distanceα-spectral radius of a connected graph,J. Inequal. Appl.2020(2020) Art# 161. · Zbl 1503.05033
[7] M. Merajuddin, S. Bhatnagar, S. Pirzada, On spectral radius and Nordhaus-Gaddum type inequalities of the generalized distance matrix of graphs, Carpathian Math. Publ., To appear. · Zbl 1499.05380
[8] S. Pirzada,An Introduction to Graph Theory, Orient BlackSwan, Hyderabad, 2012.
[9] S. Pirzada, B. A. Rather, H. A. Ganie, R. U. Shaban, On the generalized distance spectral radius of a bipartite graph,Mat. Vesnik72(2020) 327-336 · Zbl 1474.05200
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