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Register a new userA Characterization of Oriented Hypergraphic Laplacian and Adjacency Matrix Coefficients
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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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Restrictions of incidence-preserving path maps produce an oriented hypergraphic All Minors Matrix-tree Theorems for Laplacian and adjacency matrices. The images of these maps produce a locally signed graphic, incidence generalization, of cycle covers and basic figures that correspond to incidence-k-forests. When restricted to bidirected graphs the natural partial ordering of maps results in disjoint signed boolean lattices whose minor calculations correspond to principal order ideals. As an application, (1) the determinant formula of a signed graphic Laplacian is reclaimed and shown to be determined by the maximal positive-circle-free elements, and (2) spanning trees are equivalent to single-element order ideals.
An oriented hypergraph is a hypergraph where each vertex-edge incidence is given a label of $+1$ or $-1$. We define the adjacency, incidence and Laplacian matrices of an oriented hypergraph and study each of them. We extend several matrix results known for graphs and signed graphs to oriented hypergraphs. New matrix results that are not direct generalizations are also presented. Finally, we study a new family of matrices that contains walk information.
During routine state space circuit analysis of an arbitrarily connected set of nodes representing a lossless LC network, a matrix was formed that was observed to implicitly capture connectivity of the nodes in a graph similar to the conventional incidence matrix, but in a slightly different manner. This matrix has only 0, 1 or -1 as its elements. A sense of direction (of the graph formed by the nodes) is inherently encoded in the matrix because of the presence of -1. It differs from the incidence matrix because of leaving out the datum node from the matrix. Calling this matrix as forward adjacency matrix, it was found that its inverse also displays useful and interesting physical properties when a specific style of node-indexing is adopted for the nodes in the graph. The graph considered is connected but does not have any closed loop/cycle (corresponding to closed loop of inductors in a circuit) as with its presence the matrix is not invertible. Incidentally, by definition the graph being considered is a tree. The properties of the forward adjacency matrix and its inverse, along with rigorous proof, are presented.
This contribution gives an extensive study on spectra of mixed graphs via its Hermitian adjacency matrix of the second kind introduced by Mohar [21]. This matrix is indexed by the vertices of the mixed graph, and the entry corresponding to an arc from $u$ to $v$ is equal to the sixth root of unity $omega=frac{1+{bf i}sqrt{3}}{2}$ (and its symmetric entry is $overline{omega}=frac{1-{bf i}sqrt{3}}{2}$); the entry corresponding to an undirected edge is equal to 1, and 0 otherwise. The main results of this paper include the following: Some interesting properties are discovered about the characteristic polynomial of this novel matrix. Cospectral problems among mixed graphs, including mixed graphs and their underlying graphs, are studied. We give equivalent conditions for a mixed graph that shares the same spectrum of its Hermitian adjacency matrix of the second kind ($H_S$-spectrum for short) with its underlying graph. A sharp upper bound on the $H_S$-spectral radius is established and the corresponding extremal mixed graphs are identified. Operations which are called three-way switchings are discussed--they give rise to a large number of $H_S$-cospectral mixed graphs. We extract all the mixed graphs whose rank of its Hermitian adjacency matrix of the second kind ($H_S$-rank for short) is $2$ (resp. 3). Furthermore, we show that all connected mixed graphs with $H_S$-rank $2$ can be determined by their $H_S$-spectrum. However, this does not hold for all connected mixed graphs with $H_S$-rank $3$. We identify all mixed graphs whose eigenvalues of its Hermitian adjacency matrix of the second kind ($H_S$-eigenvalues for short) lie in the range $(-alpha,, alpha)$ for $alphainleft{sqrt{2},,sqrt{3},,2right}$.
Let $P_n$ and $C_n$ denote the path and cycle on $n$ vertices respectively. The dumbbell graph, denoted by $D_{p,k,q}$, is the graph obtained from two cycles $C_p$, $C_q$ and a path $P_{k+2}$ by identifying each pendant vertex of $P_{k+2}$ with a vertex of a cycle respectively. The theta graph, denoted by $Theta_{r,s,t}$, is the graph formed by joining two given vertices via three disjoint paths $P_{r}$, $P_{s}$ and $P_{t}$ respectively. In this paper, we prove that all dumbbell graphs as well as theta graphs are determined by their Laplacian spectra.
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