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

Pirzada, Shariefuddin

DML, Discrete Math. Lett. 7, 21-23 (2021).

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

Keywords:

distance matrix; generalized distance matrix; spectral radius; transmission degree

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