Preface 1. Introduction 2. Graph operations and modifications 3. Spectrum and structure 4. Characterizations by spectra 5. Structure and one eigenvalue 6. Spectral techniques 7. Laplacians 8. Additional topics 9. Applications Appendix Bibliography Index of symbols Index.

An Introduction to the Theory of Graph Spectra: Spectral techniques

A signed graph is a graph whose edges are labeled by signs. This is a bibliography of signed graphs and related mathematics. Several kinds of labelled graph have been called "signed" yet are mathematically very different. I distinguish four types: Group-signed graphs: the edge labels are elements of a 2-element group and are multiplied around a polygon (or along any walk). Among the natural generalizations are larger groups and vertex signs. Sign-colored graphs, in which the edges are labelled from a two-element set that is acted upon by the sign group: - interchanges labels, + leaves them unchanged. This is the kind of "signed graph" found in knot theory. The natural generalization is to more colors and more general groups —  or no group. Weighted graphs, in which the edge labels are the elements +1 and -1 of the integers or another additive domain. Weights behave like numbers, not signs; thus I regard work on weighted graphs as outside the scope of the bibliography — ex cept (to some extent) when the author calls the weights "signs". Labelled graphs where the labels have no structure or properties but are called "signs" for any or no reason.

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A Mathematical Bibliography of Signed and Gain Graphs and Allied Areas

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An Introduction To The Theory Of Graph Spectra

Pose graph optimization (PGO) is the problem of estimating a set of poses from pairwise relative measurements PGO is a nonconvex problem and, currently, no known technique can guarantee the computation of a global optimal solution In this paper, we show that Lagrangian duality allows computing a globally optimal solution under conditions that are satisfied in most robotics applications and enables to certify optimality of a given estimate Our first contribution is to frame planar PGO in the complex domain This makes analysis easier and allows drawing connections with existing literature on unit gain graphs The second contribution is to formulate and analyze the properties of the Lagrangian dual problem in the complex domain Our analysis shows that the duality gap is connected to the number of zero eigenvalues of the penalized pose graph matrix We prove that if this matrix has a single zero eigenvalue, then 1) the duality gap is zero, 2) the primal PGO problem has a unique solution (up to an arbitrary roto-translation), and 3) the primal solution can be computed by scaling an eigenvector of the penalized pose graph matrix The third contribution is algorithmic: We leverage duality to devise and algorithm that computes the optimal solution when the penalized matrix has a single zero eigenvalue We also propose a suboptimal variant when the zero eigenvalues are multiple Finally, we show that duality provides computational tools to verify if a given estimate (eg, computed using iterative solvers) is globally optimal We conclude the paper with an extensive numerical analysis Empirical evidence shows that, in the vast majority of cases ( $\text{100\%}$ of the tests under noise regimes of practical robotics applications), the penalized pose graph matrix has a single zero eigenvalue; hence, our approach allows computing (or verifying) the optimal solution

Planar Pose Graph Optimization: Duality, Optimal Solutions, and Verification

On the Laplacian coefficients of signed graphs

Spectral properties of complex unit gain graphs

An oriented hypergraphic approach to algebraic graph theory

An oriented hypergraph is a hypergraph where each vertex-edge incidence is given a label of +1 or 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. An oriented hypergraph is a hypergraph where each vertex-edge incidence is given a label of +1 or −1. This incidence structure can be viewed as a generalization of an oriented signed graph (22). In (18), the author and Rusnak studied several matrices associated with an oriented hypergraph. In this paper, we study the eigenvalues associated to the adjacency and Laplacian matrices of an oriented hypergraph. The paper is organized in the following manner. In Section 2, a background on oriented hypergraphs and their matrices is provided. In Section 3, the adjacency ma- trix is further investigated. Vertex-switching is shown to produce cospectral oriented hypergraphs. Also, bounds for the spectral radius and eigenvalues of the adjacency

https://journals.uwyo.edu/index.php/ela/article/download/1373/1373

Nathan Reff

Papers

Spectral properties of complex unit gain graphs

An oriented hypergraphic approach to algebraic graph theory

Spectral properties of oriented hypergraphs

Oriented gain graphs, line graphs and eigenvalues

Spectra of cycle and path families of oriented hypergraphs