On distance matrices of wheel graphs with an odd number of vertices
TL;DR: In this article, the wheel graph having n-vertices is considered, and if i and j are any two vertices of W n, define d i j := 0 if i = j 1 if i ǫ and jǫ are adjacent 2 else.
Abstract: Let W n denote the wheel graph having n-vertices. If i and j are any two vertices of W n , define d i j := 0 if i = j 1 if i and j are adjacent 2 else . Let D be the n × n matrix with ( i , j ...
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TL;DR: In this paper, it was shown that det(D) = 3 (n − 1 ) 2 n − 1 2 n−1 2 N − 1 1 2 L + 4 3 ( n− 1 ) u u u, where u is the number of vertices in the wheel graph.
4 citations
TL;DR: In this article , the rank of the 2-Steiner distance matrix of a caterpillar graph having vertices and veritices is shown to be 2N-p-1 .
Abstract: Abstract In this article, we show that the rank of the 2-Steiner distance matrix of a caterpillar graph having N N vertices and p p pendant veritices is 2 N − p − 1 2N-p-1 .
1 citations
TL;DR: In this article , it was shown that the generalized resistance is a non-negative real number for strongly connected and matrix weighted balanced directed graphs with scalar weights, where the weights are scalars.
Abstract: Let G be a connected graph with V(G)={1,…,n}. Then the resistance distance between any two vertices i and j is given by rij:=lii†+ljj†−2lij†, where lij† is the (i,j)th entry of the Moore-Penrose inverse of the Laplacian matrix of G. For the resistance matrix R:=[rij], there is an elegant formula to compute the inverse of R. This says that R−1=−12L+1τ′Rτττ′,where τ:=(τ1,…,τn)′andτi:=2−∑j∼iriji=1,…,n.A far reaching generalization of this result that gives an inverse formula for a generalized resistance matrix of a strongly connected and matrix weighted balanced directed graph is obtained in this paper. When the weights are scalars, it is shown that the generalized resistance is a non-negative real number. We also obtain a perturbation result involving resistance matrices of connected graphs and Laplacians of digraphs.
TL;DR: In this paper , the authors discuss the strong chromatic index of graphs generated by inflating some common classes of graphs and graphs derived from them, such as the cycle graph and the star graph.
Abstract: The problem of strong edge coloring discusses assigning colors to the edges of a graph such that distinct colors are assigned to any two edges which are either adjacent to each other or are adjacent to a common edge. The least number of colors required to define a strong edge coloring of a graph is called its strong chromatic index. This problem is equivalent to the problem of assigning collision-free frequencies to the links between the elements of a wireless sensor network. In this article, we discuss a novel way of generating new graphs from existing graphs. This graph construction is known as inflating a graph. We discuss the strong chromatic index of graphs generated by inflating some common classes of graphs and graphs derived from them. In particular, we consider the cycle graph, which is symmetric in nature, and graphs such as the path graph and the star graph, which are not symmetric. Further, we analyze the factors which influence the strong chromatic index of these inflated graphs.
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01 Jan 2014
TL;DR: In this article, the authors present a matrix game based on graph games, where the objective is to find the positive definite completion problem in a graph. But the game is not suitable for children.
Abstract: Preliminaries.- Incidence Matrix.- Adjacency Matrix.- Laplacian Matrix.- Cycles and Cuts.- Regular Graphs.- Line Graph of a Tree.- Algebraic Connectivity.- Distance Matrix of a Tree.- Resistance Distance.- Laplacian Eigenvalues of Threshold Graphs.- Positive Definite Completion Problem.- Matrix Games Based on Graphs.
482 citations
TL;DR: The present paper reports on the results related to the distance matrix of a graph and its spectral properties.
233 citations
TL;DR: In this article, the authors describe exactly how the coefficients δK(T) depend on the structure of a tree T. In contrast to the corresponding problem for the adjacency matrix of T, the results here are surprisingly difficult, requiring the use of a number of interesting auxiliary results.
223 citations
01 Nov 2013
TL;DR: In this paper, the spectral properties of the distance matrix of a connected graph and its spectral properties were investigated and the authors reported the results related to the distance matrices of a graph and their spectral properties.
Abstract: In 1971, Graham and Pollack established a relationship between the number of negative eigenvalues of the distance matrix and the addressing problem in data communication systems. They also proved that the determinant of the distance matrix of a tree is a function of the number of vertices only. Since then several mathematicians were interested in studying the spectral properties of the distance matrix of a connected graph. Computing the distance characteristic polynomial and its coefficients was the first research subject of interest. Thereafter, the eigenvalues attracted much more attention. In the present paper, we report on the results related to the distance matrix of a graph and its spectral properties.
212 citations
TL;DR: In this article, the determinant of the distance matrix of a weighted tree for a perturbation of D−1 was shown to be an entry-wise positive matrix, and the inertia of the tree was investigated.
129 citations