On the $D_\alpha$ spectral radius of nontransmission regular graphs
Abstract
Let $G$ be a connected graph with order $n$ and size $m$. Let $D(G)$ and $Tr(G)$ be the distance matrix and diagonal matrix with vertex transmissions of $G$, respectively. For any real $\alpha\in[0,1]$, the generalized distance matrix $D_\alpha(G)$ of $G$ is defined as $$D_\alpha(G)=\alpha Tr(G)+(1\alpha)D(G).$$ The $D_\alpha$ spectral radius of $G$ is the spectral radius of $D_\alpha(G)$, denoted by $\mu_{\alpha}(G)$. In this paper, we establish a lower bound on the difference between the maximum vertex transmission and the $D_\alpha$ spectral radius of nontransmission regular graphs, and we also characterize the extremal graphs attaining the bound.
 Publication:

arXiv eprints
 Pub Date:
 February 2024
 DOI:
 10.48550/arXiv.2402.03404
 arXiv:
 arXiv:2402.03404
 Bibcode:
 2024arXiv240203404X
 Keywords:

 Mathematics  Combinatorics