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A signed network Σ = (G , σ) with the underlying graph G = (V , E) , used as a mathematical model for analyzing social networks, has each edge in E with a weight 1 or − 1 assigned by the signature function σ. In this paper, we deal with two types of Detour Distance Laplacian (DDL) matrices for signed networks. We characterize balance in signed social networks using these matrices and we compute the DDL spectrum of certain unbalanced signed networks, as balanced signed networks behave like unsigned ones. [ABSTRACT FROM AUTHOR]

The distance, distance signless Laplacian and distance Laplacian matrix of a simple connected graph G , are denoted by D (G) , D Q (G) = D (G) + Tr (G) and D L (G) = Tr (G) − D (G) , respectively, where Tr (G) is the diagonal matrix of vertex transmission. Ger s ̌ gorin discs for any n × n square matrix A = [ a i j ] are the discs { z ∈ ℂ : | z − a i i | ≤ R i (A) } , where R i (A) = ∑ j ≠ i | a i j | , i = 1 , 2 , ... , n. The famous Ger s ̌ gorin disc theorem says that all the eigenvalues of a square matrix lie in the union of the Ger s ̌ gorin discs of that matrix. In this paper, some classes of graphs are studied for which the smallest Ger s ̌ gorin disc contains every distance and distance signless Laplacian eigenvalues except the spectral radius of the corresponding matrix. For all connected graphs, a lower bound and for trees, an upper bound of every distance signless Laplacian eigenvalues except the spectral radius is given in this paper. These bounds are comparatively better than the existing bounds. By applying these bounds, we find some infinite classes of graphs for which the smallest Ger s ̌ gorin disc contains every distance signless Laplacian eigenvalues except the spectral radius of the distance signless Laplacian matrix. For the distance Laplacian eigenvalues, we have given an upper bound and then find a condition for which the smallest Ger s ̌ gorin disc contains every distance Laplacian eigenvalue of the distance Laplacian matrix. These results give partial answers from some questions that are raised in [2]. [ABSTRACT FROM AUTHOR]

The distance matrix of a simple connected graph G is D (G) = (d i j) , where d i j 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 matrices of the same size. The distance d i j 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 the rank of (matrix) weighted Laplacian matrix associated with it. We present 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. Then we study some of the properties of the distance matrices of matrix weighted trees in connection with the Laplacian matrices, incidence matrices, and g-inverses. Finally, we derive an interlacing inequality for the eigenvalues of distance and Laplacian matrices for the case of positive definite matrix weights. [ABSTRACT FROM AUTHOR]

A partition of a square matrix A is said to be equitable if all the blocks of the partitioned matrix have constant row sums and each of the diagonal blocks are of square order. A quotient matrixQ of a square matrix A corresponding to an equitable partition is a matrix whose entries are the constant row sums of the corresponding blocks of A. A quotient matrix is a useful tool to find some eigenvalues of the matrix A. In this paper we determine some matrices whose eigenvalues are those eigenvalues of A which are not the eigenvalues of a quotient matrix of A. Using this result we find eigenvalue localization theorems for matrices having an equitable partition. In particular, we find eigenvalue localization theorems for stochastic matrices and give a suitable example to compare with the existing results. [ABSTRACT FROM AUTHOR]

The resistance matrix of a simple connected graph G is denoted by R G or simply by R and is defined by R G = (r i j) , where r i j is the resistance distance between the vertices i and j in G. In this paper, we consider distance matrix of a weighted tree and the resistance matrix of any weighted graph, where the weights are nonzero scalars. We obtain the identities for minors of the Laplacian, resistance and distance matrices, which are independent of the nonsingularity of resistance and distance matrices. While finding these we obtain the necessary and sufficient condition for the resistance matrix to be singular and the rank of it. Finally, we obtain the Moore–Penrose inverse of R, when it is singular. [ABSTRACT FROM AUTHOR]