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

In order to store a graph or digraph in a computer, we need something other than the diagram or the formal definition. This something is the adjacency matrix, a matrix of O’s and l’s. The l’s correspond to the arcs of the digraph. Certain matrix operations will be seen to correspond to digraph concepts.

Graphs and Matrices

Combinatorial Matrix Theory

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

Graphs and matrices

On weighted directed graphs

Abstract Let G be a graph on n vertices and A(G), L(G), and |L|(G) be the adjacency matrix, Laplacian matrix and signless Laplacian matrix of G, respectively. The paper is essentially a survey of known results about the spectra of the adjacency, Laplacian and signless Laplacian matrix of graphs resulting from various graph operations with special emphasis on corona and graph products. In most cases, we have described the eigenvalues of the resulting graphs along with an explicit description of the structure of the corresponding eigenvectors.

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Spectra of Graphs Resulting from Various Graph Operations and Products: a Survey

A reciprocal eigenvalue property for unicyclic weighted directed graphs with weights from {±1,±i}

The notion of a 3-coloured digraph introduced in Bapat et al. [R.B. Bapat, D. Kalita, and S. Pati, On weighted directed graphs, Linear Algebra Appl. (2011), In press, doi:10.1016/j.laa.2011.06.035] is a generalization of the mixed graphs. In this article, we study the adjacency and the Laplacian spectra of 3-coloured digraphs. Our main objective is to investigate whether the adjacency (resp. Laplacian) spectrum of a 3-coloured digraph can be realized as a subset of the adjacency (resp. Laplacian) spectrum of a suitable undirected graph. In order to achieve this, some graph operations similar to that in Fan [Y.Z. Fan, On eigenvectors of mixed graphs with exactly one nonsingular cycle, Czech. Math. J. 57 (2007), pp. 1215–1222] are introduced. Using these graph operations, we show that for a connected 3-coloured digraph on n vertices, there exists a mixed graph on 2n vertices whose adjacency and Laplacian eigenvalues are precisely those of the 3-coloured digraph with doubled multiplicities. We also show that...

Debajit Kalita

Papers

On weighted directed graphs

Spectra of Graphs Resulting from Various Graph Operations and Products: a Survey

A reciprocal eigenvalue property for unicyclic weighted directed graphs with weights from {±1,±i}

On the spectrum of 3-coloured digraphs

Convex and quasiconvex functions on trees and their applications