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

Assignment problems deal with the question how to assign n items (eg jobs) to n machines (or workers) in the best possible way They consist of two components: the assignment as underlying combinatorial structure and an objective function modeling the ”best way”

/pdf/linear-assignment-problems-and-extensions-37goib9knk.pdf

Linear Assignment Problems and Extensions

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.

/pdf/a-mathematical-bibliography-of-signed-and-gain-graphs-and-3a49f10hs2.pdf

A Mathematical Bibliography of Signed and Gain Graphs and Allied Areas

https://www.polyu.edu.hk/ama/staff/new/qilq/DQW.pdf

M-tensors and nonsingular M-tensors

A general product of tensors with applications

In this paper, we study some extremal problems of three kinds of spectral radii of $$k$$k-uniform hypergraphs (the adjacency spectral radius, the signless Laplacian spectral radius and the incidence $$Q$$Q-spectral radius). We call a connected and acyclic $$k$$k-uniform hypergraph a supertree. We introduce the operation of "moving edges" for hypergraphs, together with the two special cases of this operation: the edge-releasing operation and the total grafting operation. By studying the perturbation of these kinds of spectral radii of hypergraphs under these operations, we prove that for all these three kinds of spectral radii, the hyperstar $$\mathcal {S}_{n,k}$$Sn,k attains uniquely the maximum spectral radius among all $$k$$k-uniform supertrees on $$n$$n vertices. We also determine the unique $$k$$k-uniform supertree on $$n$$n vertices with the second largest spectral radius (for these three kinds of spectral radii). We also prove that for all these three kinds of spectral radii, the loose path $$\mathcal {P}_{n,k}$$Pn,k attains uniquely the minimum spectral radius among all $$k$$k-th power hypertrees of $$n$$n vertices. Some bounds on the incidence $$Q$$Q-spectral radius are given. The relation between the incidence $$Q$$Q-spectral radius and the spectral radius of the matrix product of the incidence matrix and its transpose is discussed.

The extremal spectral radii of $$k$$k-uniform supertrees

https://www.polyu.edu.hk/ama/staff/new/qilq/1-s2.0-S0024379513005429-main.pdf

Cored Hypergraphs, Power Hypergraphs and Their Laplacian H-Eigenvalues ⁄

On some properties of the determinants of tensors

A -uniform hypergraph is called odd-bipartite, if is even and there exists some proper subset of such that each edge of contains odd number of vertices in . Odd-bipartite hypergraphs are generalizations of the ordinary bipartite graphs. We study the spectral properties of the connected odd-bipartite hypergraphs. We prove that the Laplacian H-spectrum and signless Laplacian H-spectrum of a connected -uniform hypergraph are equal if and only if is even and is odd-bipartite. We further give several spectral characterizations of the connected odd-bipartite hypergraphs. We also give a characterization for a connected -uniform hypergraph whose Laplacian spectral radius and signless Laplacian spectral radius are equal; thus, provide an answer to a question raised by L. Qi. By showing that the Cartesian product of two odd-bipartite -uniform hypergraphs is still odd-bipartite, we determine that the Laplacian spectral radius of is the sum of the Laplacian spectral radii of and , when and are both connected odd-bipa...

Jia-Yu Shao

Papers

A general product of tensors with applications

The extremal spectral radii of $$k$$k-uniform supertrees

Cored Hypergraphs, Power Hypergraphs and Their Laplacian H-Eigenvalues ⁄

On some properties of the determinants of tensors

Some spectral properties and characterizations of connected odd-bipartite uniform hypergraphs