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Register a new userOn distance and Laplacian matrices of trees with matrix weights
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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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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.
In this paper, we use a new and correct method to determine the $n$-vertex $k$-trees with the first three largest signless Laplacian indices.
Geodesic distance, sometimes called shortest path length, has proven useful in a great variety of applications, such as information retrieval on networks including treelike networked models. Here, our goal is to analytically determine the exact solutions to geodesic distances on two different families of growth trees which are recursively created upon an arbitrary tree $mathcal{T}$ using two types of well-known operations, first-order subdivision and ($1,m$)-star-fractal operation. Different from commonly-used methods, for instance, spectral techniques, for addressing such a problem on growth trees using a single edge as seed in the literature, we propose a novel method for deriving closed-form solutions on the presented trees completely. Meanwhile, our technique is more general and convenient to implement compared to those previous methods mainly because there are not complicated calculations needed. In addition, the closed-form expression of mean first-passage time ($MFPT$) for random walk on each member in tree families is also readily obtained according to connection of our obtained results to effective resistance of corresponding electric networks. The results suggest that the two topological operations above are sharply different from each other due to $MFPT$ for random walks, and, however, have likely to show the similar performance, at least, on geodesic distance.
The distance energy of a simple connected graph $G$ is defined as the sum of absolute values of its distance eigenvalues. In this paper, we mainly give a positive answer to a conjecture of distance energy of clique trees proposed by Lin, Liu and Lu [H.~Q.~ Lin, R.~F.~Liu, X.~W.~Lu, The inertia and energy of the distance matrix of a connected graph, {it Linear Algebra Appl.,} 467 (2015), 29-39.]
An oriented hypergraph is an oriented incidence structure that generalizes and unifies graph and hypergraph theoretic results by examining its locally signed graphic substructure. In this paper we obtain a combinatorial characterization of the coefficients of the characteristic polynomials of oriented hypergraphic Laplacian and adjacency matrices via a signed hypergraphic generalization of basic figures of graphs. Additionally, we provide bounds on the determinant and permanent of the Laplacian matrix, characterize the oriented hypergraphs in which the upper bound is sharp, and demonstrate that the lower bound is never achieved.
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