Geodesic
Some properties of generalized distance eigenvalues of graphs
Czechoslovak Mathematical Journal, Tome 74 (2024) no. 1, pp. 1-15
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Let $G$ be a simple connected graph with vertex set $V(G)=\{v_1,v_2,\dots ,v_n \}$ and edge set $E(G)$, and let $d_{v_{i}}$ be the degree of the vertex $v_i$. Let $D(G)$ be the distance matrix and let $T_r(G)$ be the diagonal matrix of the vertex transmissions of $G$. The generalized distance matrix of $G$ is defined as $D_\alpha (G)=\alpha T_r(G)+(1-\alpha )D(G)$, where $0\leq \alpha \leq 1$. Let $\lambda _1(D_{\alpha }(G))\geq \lambda _2(D_{\alpha }(G)) \geq \ldots \geq \lambda _n(D_{\alpha }(G))$ be the generalized distance eigenvalues of $G$, and let $k$ be an integer with $1\leq k\leq n$. We denote by $S_{k}(D_{\alpha }(G))=\lambda _{1}(D_{\alpha }(G)) +\lambda _{2}(D_{\alpha }(G))+\ldots +\lambda _{k}(D_{\alpha }(G))$ the sum of the $k$ largest generalized distance eigenvalues. The generalized distance spread of a graph $G$ is defined as $D_{\alpha }S(G)=\lambda _{1}(D_{\alpha }(G))-\lambda _{n}(D_{\alpha }(G))$. We obtain some bounds on $S_k((D_{\alpha }(G)))$ and $D_{\alpha }S(G)$ of graph $G$, respectively.
Let $G$ be a simple connected graph with vertex set $V(G)=\{v_1,v_2,\dots ,v_n \}$ and edge set $E(G)$, and let $d_{v_{i}}$ be the degree of the vertex $v_i$. Let $D(G)$ be the distance matrix and let $T_r(G)$ be the diagonal matrix of the vertex transmissions of $G$. The generalized distance matrix of $G$ is defined as $D_\alpha (G)=\alpha T_r(G)+(1-\alpha )D(G)$, where $0\leq \alpha \leq 1$. Let $\lambda _1(D_{\alpha }(G))\geq \lambda _2(D_{\alpha }(G)) \geq \ldots \geq \lambda _n(D_{\alpha }(G))$ be the generalized distance eigenvalues of $G$, and let $k$ be an integer with $1\leq k\leq n$. We denote by $S_{k}(D_{\alpha }(G))=\lambda _{1}(D_{\alpha }(G)) +\lambda _{2}(D_{\alpha }(G))+\ldots +\lambda _{k}(D_{\alpha }(G))$ the sum of the $k$ largest generalized distance eigenvalues. The generalized distance spread of a graph $G$ is defined as $D_{\alpha }S(G)=\lambda _{1}(D_{\alpha }(G))-\lambda _{n}(D_{\alpha }(G))$. We obtain some bounds on $S_k((D_{\alpha }(G)))$ and $D_{\alpha }S(G)$ of graph $G$, respectively.
DOI : 10.21136/CMJ.2023.0136-21
Classification : 05C12, 05C50, 15A18
Keywords: graph; generalized distance matrix; generalized distance eigenvalue; generalized distance spread 
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Ma, Yuzheng; Shao, Yanling. Some properties of generalized distance eigenvalues of graphs. Czechoslovak Mathematical Journal, Tome 74 (2024) no. 1, pp. 1-15. doi: 10.21136/CMJ.2023.0136-21

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