On the Largest Distance (Signless Laplacian) Eigenvalue of Non-transmission-regular Graphs

Abstract

Let $G=(V(G),E(G))$ be a $k$-connected graph with $n$ vertices and $m$ edges. Let $D(G)$ be the distance matrix of $G$. Suppose $\lambda_1(D)\geq \cdots \geq \lambda_n(D)$ are the $D$-eigenvalues of $G$. The transmission of $v_i \in V(G)$, denoted by $Tr_G(v_i)$ is defined to be the sum of distances from $v_i$ to all other vertices of $G$, i.e., the row sum $D_{i}(G)$ of $D(G)$ indexed by vertex $v_i$ and suppose that $D_1(G)\geq \cdots \geq D_n(G)$. The $Wiener~ index$ of $G$ denoted by $W(G)$ is given by $W(G)=\frac{1}{2}\sum_{i=1}^{n}D_i(G)$. Let $Tr(G)$ be the $n\times n$ diagonal matrix with its $(i,i)$-entry equal to $TrG(v_i)$. The distance signless Laplacian matrix of $G$ is defined as $D^Q(G)=Tr(G)+D(G)$ and its spectral radius is denoted by $\rho_1(D^Q(G))$ or $\rho_1$. A connected graph $G$ is said to be $t$-transmission-regular if $Tr_G(v_i) =t$ for every vertex $v_i\in V(G)$, otherwise, non-transmission-regular. In this paper, we respectively estimate $D_1(G)-\lambda_1(G)$ and $2D_1(G)-\rho_1(G)$ for a $k$-connected non-transmission-regular graph in different ways and compare these obtained results. And we conjecture that $D_1(G)-\lambda_1(G)>\frac{1}{n+1}$. Moreover, we show that the conjecture is valid for trees.

On the distance $α$-spectral radius of a connected graph

“…As µ α (G) ≤ T max (G) with equality if and only if G is transmission regular. Recently, Liu et al [13] showed that…”

Bounds for Generalized Distance Spectral Radius and the Entries of the Principal Eigenvector

“…The matrix D L (G) = T r(G) − D(G) is called the distance Laplacian matrix of G, while the matrix D Q (G) = T r(G) + D(G) is called the distance signless Laplacian matrix of G. The spectral properties of D(G), D L (G) and D Q (G) have attracted much more attention of the researchers and a large number of papers have been published regarding their spectral properties, like spectral radius, second largest eigenvalue, smallest eigenvalue etc. For some recent works we refer to [3,8,4,15,17,18,19,20,21,22,25,26,27] and the references therein.…”

“…Another problem worth to mention is to characterize the extremal graphs for the spectral radius of a graph matrix, in some classes of graphs. For some recent works we refer to [15,19,20,22,25,26,27] and the references therein.…”

A proof of a conjecture on the distance spectral radius and maximum transmission of graphs

“…In [3], Liu, Shu and Xue studied the bounds of D max (G) − λ 1 (G) for connected nontransmission-regular graphs, and posed the following conjecture.…”