Distance-Regular Graphs

A spectral graph theory is a theory in which graphs are studied by means of eigenvalues of a matrix �� which is in a prescribed way defined for any graph. This theory is called �� -theory. We outline a spectral theory of graphs based on the signless Laplacians �� and compare it with other spectral theories, in particular with those based on the adjacency matrix �� and the Laplacian �� . The �� -theory can be composed using various connections to other theories: equivalency with �� -theory and �� -theory for regular graphs, or with �� -theory for bipartite graphs, general analogies with �� -theory and analogies with �� -theory via line graphs and subdivision graphs. We present results on graph operations, inequalities for eigenvalues and reconstruction problems.

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Towards a spectral theory of graphs based on the signless Laplacian, I

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

Towards a spectral theory of graphs based on the signless Laplacian, II

On the distance spectrum of graphs

On the two largest Q-eigenvalues of graphs

On the spectral characterizations of ∞-graphs

Signed graphs are graphs whose edges get a sign +1 or −1 (the signature). Signed graphs can be studied by means of graph matrices extended to signed graphs in a natural way. Recently, the spectra of signed graphs have attracted much attention from graph spectra specialists. One motivation is that the spectral theory of signed graphs elegantly generalizes the spectral theories of unsigned graphs. On the other hand, unsigned graphs do not disappear completely, since their role can be taken by the special case of balanced signed graphs.
Therefore, spectral problems defined and studied for unsigned graphs can be considered in terms of signed graphs, and sometimes such generalization shows nice properties which cannot be appreciated in terms of (unsigned) graphs. Here, we survey some general results on the adjacency spectra of signed graphs, and we consider some spectral problems which are inspired from the spectral theory of (unsigned) graphs.

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Jianfeng Wang

Papers

On the distance spectrum of graphs

On the two largest Q-eigenvalues of graphs

On the spectral characterizations of ∞-graphs

Open problems in the spectral theory of signed graphs

On the spectral radius of unicyclic graphs with prescribed degree sequence