The generalized distance matrix of digraphs

Abstract

Let $D(G)$ and $D^Q(G)=Diag(Tr)+D(G)$ be the distance matrix and distance signless Laplacian matrix of a simple strongly connected digraph $G$, respectively, where $Diag(Tr)=\text{diag}(D_1,D_2,$ $\dots ,D_n)$ be the diagonal matrix with vertex transmissions of the digraph $G$. To track the gradual change of $D(G)$ into $D^Q(G)$, in this paper, we propose to study the convex combinations of $D(G)$ and $Diag(Tr)$ defined by $$D_{\alpha }(G)=\alpha Diag(Tr)+(1-\alpha )D(G),0\le \alpha \le 1.$$ This study reduces to merging the distance spectral and distance signless Laplacian spectral theories. The eigenvalue with the largest modulus of $D_{\alpha }(G)$ is called the $D_{\alpha }$ spectral radius of $G$, denoted by ${\mu }_{\alpha }(G)$. We determine the digraph which attains the maximum (or minimum) $D_{\alpha }$ spectral radius among all strongly connected digraphs. Moreover, we also determine the digraphs which attain the minimum $D_{\alpha }$ spectral radius among all strongly connected digraphs with given parameters such as dichromatic number, vertex connectivity or arc connectivity.

14 pages, 0 figure. arXiv admin note: substantial text overlap with arXiv:1810.11669