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Register a new userSpectral properties of the exponential distance matrix
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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.
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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.
We realize many sharp spectral bounds of the spectral radius of a nonnegative square matrix $C$ by using the largest real eigenvalues of suitable matrices of smaller sizes related to $C$ that are very easy to find. As applications, we give a sharp upper bound of the spectral radius of $C$ expressed by the sum of entries, the largest off-diagonal entry $f$ and the largest diagonal entry $d$ in $C$. We also give a new class of sharp lower bounds of the spectral radius of $C$ expressed by the above $d$ and $f$, the least row-sum $r_n$ and the $t$-th largest row-sum $r_t$ in $C$ satisfying $0<r_n-(n-t-1)f-dleq r_t-(n-t)f$, where $n$ is the size of $C$.
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.
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 $2N-p-1$.
The emph{distance matrix} of a simple connected graph $G$ is $D(G)=(d_{ij})$, where $d_{ij}$ is the distance between the vertices $i$ and $j$ in $G$. We consider a weighted tree $T$ on $n$ vertices with edge weights are square matrix of same size. The distance $d_{ij}$ between the vertices $i$ and $j$ is the sum of the weight matrices of the edges in the unique path from $i$ to $j$. In this article we establish a characterization for the trees in terms of rank of (matrix) weighted Laplacian matrix associated with it. Then we establish a necessary and sufficient condition for the distance matrix $D$, with matrix weights, to be invertible and the formula for the inverse of $D$, if it exists. Also we study some of the properties of the distance matrices of matrix weighted trees in connection with the Laplacian matrices, g-inverses and eigenvalues.
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