## 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.

### MSC:

 05C12 Distance in graphs 05C50 Graphs and linear algebra (matrices, eigenvalues, etc.) 15A18 Eigenvalues, singular values, and eigenvectors
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### References:

 [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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