Some properties of generalized distance eigenvalues of graphs

Ma, Yuzheng, and Shao, Yan Ling. "Some properties of generalized distance eigenvalues of graphs." Czechoslovak Mathematical Journal 74.1 (2024): 1-15. <http://eudml.org/doc/299228>.

@article{Ma2024,
abstract = {Let $G$ be a simple connected graph with vertex set $V(G)=\lbrace v_1,v_2,\dots ,v_n \rbrace $ 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 $0\le \alpha \le 1$. Let $\lambda _1(D_\{\alpha \}(G))\ge \lambda _2(D_\{\alpha \}(G)) \ge \ldots \ge \lambda _n(D_\{\alpha \}(G))$ be the generalized distance eigenvalues of $G$, and let $k$ be an integer with $1\le k\le n$. 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.},
author = {Ma, Yuzheng, Shao, Yan Ling},
journal = {Czechoslovak Mathematical Journal},
keywords = {graph; generalized distance matrix; generalized distance eigenvalue; generalized distance spread},
language = {eng},
number = {1},
pages = {1-15},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Some properties of generalized distance eigenvalues of graphs},
url = {http://eudml.org/doc/299228},
volume = {74},
year = {2024},
}

TY - JOUR
AU - Ma, Yuzheng
AU - Shao, Yan Ling
TI - Some properties of generalized distance eigenvalues of graphs
JO - Czechoslovak Mathematical Journal
PY - 2024
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 74
IS - 1
SP - 1
EP - 15
AB - Let $G$ be a simple connected graph with vertex set $V(G)=\lbrace v_1,v_2,\dots ,v_n \rbrace $ 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 $0\le \alpha \le 1$. Let $\lambda _1(D_{\alpha }(G))\ge \lambda _2(D_{\alpha }(G)) \ge \ldots \ge \lambda _n(D_{\alpha }(G))$ be the generalized distance eigenvalues of $G$, and let $k$ be an integer with $1\le k\le n$. 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.
LA - eng
KW - graph; generalized distance matrix; generalized distance eigenvalue; generalized distance spread
UR - http://eudml.org/doc/299228
ER -