On spectral radius and Nordhaus-Gaddum type inequalities of the generalized distance matrix of graphs
| Autor: | M. Merajuddin, S. Bhatnagar, S. Pirzada |
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| Jazyk: | English<br />Ukrainian |
| Rok vydání: | 2022 |
| Předmět: | |
| Zdroj: | Karpatsʹkì Matematičnì Publìkacìï, Vol 14, Iss 1, Pp 185-193 (2022) |
| Druh dokumentu: | article |
| ISSN: | 2075-9827 2313-0210 |
| DOI: | 10.15330/cmp.14.1.185-193 |
| Popis: | 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 $D_{\alpha}(G)=\alpha ~Tr(G)+(1-\alpha)~D(G)$, where $0\leq \alpha \leq 1$. If $\rho_1 \geq \rho_2 \geq \dots \geq \rho_n$ are the eigenvalues of $D_{\alpha}(G)$, the largest eigenvalue $\rho_1$ (or $\rho_{\alpha}(G)$) is called the spectral radius of the generalized distance matrix $D_{\alpha}(G)$. The generalized distance energy is defined as $E^{D_{\alpha}}(G)=\sum_{i=1}^{n}\left|\rho_i -\frac{2\alpha W(G)}{n}\right|$, where $W(G)$ is the Wiener index of $G$. In this paper, we obtain the bounds for the spectral radius $\rho_{\alpha}(G)$ and the generalized distance energy of $G$ involving Wiener index. We derive the Nordhaus-Gaddum type inequalities for the spectral radius and the generalized distance energy of $G$. |
| Databáze: | Directory of Open Access Journals |
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