1,456 research outputs found

    Spectral Properties of Oriented Hypergraphs

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    An oriented hypergraph is a hypergraph where each vertex-edge incidence is given a label of +1+1 or βˆ’1-1. The adjacency and Laplacian eigenvalues of an oriented hypergraph are studied. Eigenvalue bounds for both the adjacency and Laplacian matrices of an oriented hypergraph which depend on structural parameters of the oriented hypergraph are found. An oriented hypergraph and its incidence dual are shown to have the same nonzero Laplacian eigenvalues. A family of oriented hypergraphs with uniformally labeled incidences is also studied. This family provides a hypergraphic generalization of the signless Laplacian of a graph and also suggests a natural way to define the adjacency and Laplacian matrices of a hypergraph. Some results presented generalize both graph and signed graph results to a hypergraphic setting.Comment: For the published version of the article see http://repository.uwyo.edu/ela/vol27/iss1/24

    Spectral Properties of Complex Unit Gain Graphs

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    A complex unit gain graph is a graph where each orientation of an edge is given a complex unit, which is the inverse of the complex unit assigned to the opposite orientation. We extend some fundamental concepts from spectral graph theory to complex unit gain graphs. We define the adjacency, incidence and Laplacian matrices, and study each of them. The main results of the paper are eigenvalue bounds for the adjacency and Laplacian matrices.Comment: 13 pages, 1 figure, to appear in Linear Algebra App

    Spectral convergence of non-compact quasi-one-dimensional spaces

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    We consider a family of non-compact manifolds X_\eps (``graph-like manifolds'') approaching a metric graph X0X_0 and establish convergence results of the related natural operators, namely the (Neumann) Laplacian \laplacian {X_\eps} and the generalised Neumann (Kirchhoff) Laplacian \laplacian {X_0} on the metric graph. In particular, we show the norm convergence of the resolvents, spectral projections and eigenfunctions. As a consequence, the essential and the discrete spectrum converge as well. Neither the manifolds nor the metric graph need to be compact, we only need some natural uniformity assumptions. We provide examples of manifolds having spectral gaps in the essential spectrum, discrete eigenvalues in the gaps or even manifolds approaching a fractal spectrum. The convergence results will be given in a completely abstract setting dealing with operators acting in different spaces, applicable also in other geometric situations.Comment: some references added, still 36 pages, 4 figure

    Unsolved Problems in Spectral Graph Theory

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    Spectral graph theory is a captivating area of graph theory that employs the eigenvalues and eigenvectors of matrices associated with graphs to study them. In this paper, we present a collection of 2020 topics in spectral graph theory, covering a range of open problems and conjectures. Our focus is primarily on the adjacency matrix of graphs, and for each topic, we provide a brief historical overview.Comment: v3, 30 pages, 1 figure, include comments from Clive Elphick, Xiaofeng Gu, William Linz, and Dragan Stevanovi\'c, respectively. Thanks! This paper will be published in Operations Research Transaction

    The inertia of weighted unicyclic graphs

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    Let GwG_w be a weighted graph. The \textit{inertia} of GwG_w is the triple In(Gw)=(i+(Gw),iβˆ’(Gw),In(G_w)=\big(i_+(G_w),i_-(G_w), i0(Gw)) i_0(G_w)\big), where i+(Gw),iβˆ’(Gw),i0(Gw)i_+(G_w),i_-(G_w),i_0(G_w) are the number of the positive, negative and zero eigenvalues of the adjacency matrix A(Gw)A(G_w) of GwG_w including their multiplicities, respectively. i+(Gw)i_+(G_w), iβˆ’(Gw)i_-(G_w) is called the \textit{positive, negative index of inertia} of GwG_w, respectively. In this paper we present a lower bound for the positive, negative index of weighted unicyclic graphs of order nn with fixed girth and characterize all weighted unicyclic graphs attaining this lower bound. Moreover, we characterize the weighted unicyclic graphs of order nn with two positive, two negative and at least nβˆ’6n-6 zero eigenvalues, respectively.Comment: 23 pages, 8figure
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