For the list object, introduced in Chapter 5, it was shown that each data element contains at most one predecessor element and one successor element. Therefore, for any given data element or node in the list structure, we can talk in terms of a next element and a previous element.

Graphs and Digraphs

https://orion.math.iastate.edu/lhogben/research/FallatHogbenMinRank07.pdf

The minimum rank of symmetric matrices described by a graph: A survey☆

Distance spectra of graphs: A survey

In 1971, Graham and Pollack established a relationship between the number of negative eigenvalues of the distance matrix and the addressing problem in data communication systems. They also proved that the determinant of the distance matrix of a tree is a function of the number of vertices only. Since then several mathematicians were interested in studying the spectral properties of the distance matrix of a connected graph. Computing the distance characteristic polynomial and its coefficients was the first research subject of interest. Thereafter, the eigenvalues attracted much more attention. In the present paper, we report on the results related to the distance matrix of a graph and its spectral properties.

Distance Spectra of Graphs: A Survey

The article gives a thorough introduction to spectra of digraphs via its Hermitian adjacency matrix. This matrix is indexed by the vertices of the digraph, and the entry corresponding to an arc from x to y is equal to the complex unity i (and its symmetric entry is −i) if the reverse arc yx is not present. We also allow arcs in both directions and unoriented edges, in which case we use 1 as the entry. This allows to use the definition also for mixed graphs. This matrix has many nice properties; it has real eigenvalues and the interlacing theorem holds for a digraph and its induced subdigraphs. Besides covering the basic properties, we discuss many differences from the properties of eigenvalues of undirected graphs and develop basic theory. The main novel results include the following. Several surprising facts are discovered about the spectral radius; some consequences of the interlacing property are obtained; operations that preserve the spectrum are discussed—they give rise to a large number of cospectral digraphs; for every 0≤α≤3, all digraphs whose spectrum is contained in the interval (−α,α) are determined.

Hermitian Adjacency Matrix of Digraphs and Mixed Graphs

http://www.math.udel.edu/~cioaba/JCTB09.pdf

Matchings in regular graphs from eigenvalues

http://www.math.wsu.edu/faculty/tsat/files/BIRSgraph.pdf

Skew-adjacency matrices of graphs

The spread of the spectrum of a graph

Let G be a connected graph. Thispaper s extreme entriesof the principal eigenvector x of G, the unique positive unit eigenvector corresponding to the greatest eigenvalue λ1 of the adjacency matrix of G.I fG hasmaximum degree ∆, the greates t entry xmax of x isat mos t 1/ 1+ λ 2 /∆. This improves a result of Papendieck and Recht. The least entry xmin of x aswell asthe principal ratio xmax/xmin are studied. It is conjectured that for connected graphs of order n ≥ 3, the principal ratio isalwaysattained by one of the lollipop graphsobtained by attaching a path graph to a vertex of a complete graph.

/pdf/principal-eigenvectors-of-irregular-graphs-2dlcl3ddlt.pdf

Principal eigenvectors of irregular graphs

http://www.math.udel.edu/~cioaba/MatchEig_Oct17.pdf

David A. Gregory

Papers

Matchings in regular graphs from eigenvalues

Skew-adjacency matrices of graphs

The spread of the spectrum of a graph

Principal eigenvectors of irregular graphs

Large matchings from eigenvalues