Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userDistance matrix correlation spectrum of graphs
148
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Let $G$ be a simple, connected graph, $mathcal{D}(G)$ be the distance matrix of $G$, and $Tr(G)$ be the diagonal matrix of vertex transmissions of $G$. The distance Laplacian matrix and distance signless Laplacian matrix of $G$ are defined by $mathcal{L}(G) = Tr(G)-mathcal{D}(G)$ and $mathcal{Q}(G) = Tr(G)+mathcal{D}(G)$, respectively. The eigenvalues of $mathcal{D}(G)$, $mathcal{L}(G)$ and $mathcal{Q}(G)$ is called the $mathcal{D}-$spectrum, $mathcal{L}-$spectrum and $mathcal{Q}-$spectrum, respectively. The generalized distance matrix of $G$ is defined as $mathcal{D}_{alpha}(G)=alpha Tr(G)+(1-alpha)mathcal{D}(G),~0leqalphaleq1$, and the generalized distance spectral radius of $G$ is the largest eigenvalue of $mathcal{D}_{alpha}(G)$. In this paper, we give a complete description of the $mathcal{D}-$spectrum, $mathcal{L}-$spectrum and $mathcal{Q}-$spectrum of some graphs obtained by operations. In addition, we present some new upper and lower bounds on the generalized distance spectral radius of $G$ and of its line graph $L(G)$, based on other graph-theoretic parameters, and characterize the extremal graphs. Finally, we study the generalized distance spectrum of some composite graphs.
rate research
Read More
For a connected graph $G:=(V,E)$, the Steiner distance $d_G(X)$ among a set of vertices $X$ is the minimum size among all the connected subgraphs of $G$ whose vertex set contains $X$. The $k-$Steiner distance matrix $D_k(G)$ of $G$ is a matrix whose rows and columns are indexed by $k-$subsets of $V$. For $k$-subsets $X_1$ and $X_2$, the $(X_1,X_2)-$entry of $D_k(G)$ is $d_G(X_1 cup X_2)$. In this paper, we show that the rank of $2-$Steiner distance matrix of a caterpillar graph on $N$ vertices and with $p$ pendant veritices is $2N-p-1$.
For a simple, undirected and connected graph $G$, $D_{alpha}(G) = alpha Tr(G) + (1-alpha) D(G)$ is called the $alpha$-distance matrix of $G$, where $alphain [0,1]$, $D(G)$ is the distance matrix of $G$, and $Tr(G)$ is the vertex transmission diagonal matrix of $G$. Recently, the $alpha$-distance energy of $G$ was defined based on the spectra of $D_{alpha}(G)$. In this paper, we define the $alpha$-distance Estrada index of $G$ in terms of the eigenvalues of $D_{alpha}(G)$. And we give some bounds on the spectral radius of $D_{alpha}(G)$, $alpha$-distance energy and $alpha$-distance Estrada index of $G$.
For a connected graph $G$ on $n$ vertices, recall that the distance signless Laplacian matrix of $G$ is defined to be $mathcal{Q}(G)=Tr(G)+mathcal{D}(G)$, where $mathcal{D}(G)$ is the distance matrix, $Tr(G)=diag(D_1, D_2, ldots, D_n)$ and $D_{i}$ is the row sum of $mathcal{D}(G)$ corresponding to vertex $v_{i}$. Denote by $rho^{mathcal{D}}(G),$ $rho_{min}^{mathcal{D}}(G)$ the largest eigenvalue and the least eigenvalue of $mathcal{D}(G)$, respectively. And denote by $q^{mathcal{D}}(G)$, $q_{min}^{mathcal{D}}(G)$ the largest eigenvalue and the least eigenvalue of $mathcal{Q}(G)$, respectively. The distance spread of a graph $G$ is defined as $S_{mathcal{D}}(G)=rho^{mathcal{D}}(G)- rho_{min}^{mathcal{D}}(G)$, and the distance signless Laplacian spread of a graph $G$ is defined as $S_{mathcal{Q}}(G)=q^{mathcal{D}}(G)-q_{min}^{mathcal{D}}(G)$. In this paper, we point out an error in the result of Theorem 2.4 in Distance spectral spread of a graph [G.L. Yu, et al, Discrete Applied Mathematics. 160 (2012) 2474--2478] and rectify it. As well, we obtain some lower bounds on ddistance signless Laplacian spread of a graph.
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 $Diag(Tr)=textrm{diag}(D_1,D_2,$ $ldots,D_n)$ 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 $$D_alpha(G)=alpha Diag(Tr)+(1-alpha)D(G), 0leq alphaleq1.$$ 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.
Given a graph $G$, the exponential distance matrix is defined entry-wise by letting the $(u,v)$-entry be $q^{text{dist}(u,v)}$, where $text{dist}(u,v)$ is the distance between the vertices $u$ and $v$ with the convention that if vertices are in different components, then $q^{text{dist}(u,v)}=0$. In this paper, we will establish several properties of the characteristic polynomial (spectrum) for this matrix, give some families of graphs which are uniquely determined by their spectrum, and produce cospectral constructions.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
