Bounds on separated pairs of subgraphs, eigenvalues and related polynomials

We give a bound on the sizes of two sets of vertices at a given minimum distance (a separated pair of subgraphs) in a graph in terms of polynomials and the spectrum of the graph. We find properties of the polynomial optimizing the bound. Explicit bounds on the number of vertices at maximal distance and distance two from a given vertex, and on the size of two equally large sets at maximal distance are given, and we find graphs for which the bounds are tight.Graphs;Eigenvalues;Polynomials;mathematics

Bounding the diameter and the mean distance of a graph from its eigenvalues: Laplacian versus adjacency matrix methods

AbstractRecently, several results bounding above the diameter and/or the mean distance of a graph from its eigenvalues have been presented. They use the eigenvalues of either the adjacency or the Laplacian matrix of the graph. The main object of this paper is to compare both methods. As expected, they are equivalent for regular graphs. However, the situation is different for nonregular graphs: While no method has a definite advantage when bounding above the diameter, the use of the Laplacian matrix seems better when dealing with the mean distance. This last statement follows from improved bounds on the mean distance obtained in the paper

Cutoff on Graphs and the Sarnak-Xue Density of Eigenvalues

It was recently shown by Lubetzky and Peres (2016) and by Sardari (2018) that Ramanujan graphs, i.e., graphs with the optimal spectrum, exhibit cutoff of the simple random walk in optimal time and have optimal almost-diameter. We prove that this spectral condition can be replaced by a weaker condition, the Sarnak-Xue density of eigenvalues property, to deduce similar results. We show that a family of Schreier graphs of the $SL_{2}\left(\mathbb{F}_{t}\right)$-action on the projective line satisfies the Sarnak-Xue density condition, and hence exhibit the desired properties. To the best of our knowledge, this is the first known example of optimal cutoff and almost-diameter on an explicit family of graphs that are neither random nor Ramanujan

The spectra of Manhattan street networks

The multidimensional Manhattan street networks constitute a family of digraphs with many interesting properties, such as vertex symmetry (in fact they are Cayley digraphs), easy routing, Hamiltonicity, and modular structure. From the known structural properties of these digraphs, we determine their spectra, which always contain the spectra of hypercubes. In particular, in the standard (two-dimensional) case it is shown that their line digraph structure imposes the presence of the zero eigenvalue with a large multiplicity

The spectra of Manhattan street networks

AbstractThe multidimensional Manhattan street networks constitute a family of digraphs with many interesting properties, such as vertex symmetry (in fact they are Cayley digraphs), easy routing, Hamiltonicity, and modular structure. From the known structural properties of these digraphs, we determine their spectra, which always contain the spectra of hypercubes. In particular, in the standard (two-dimensional) case it is shown that their line digraph structure imposes the presence of the zero eigenvalue with a large multiplicity

The alternating and adjacency polynomials, and their relation with the spectra and diameters of graphs

Let Γ be a graph on n vertices, adjacency matrix A, and distinct eigenvalues λ > λ_1 > λ_2 > · · · > λ_d. For every k = 0,1, . . . ,d −1, the k-alternating polynomial P_k is defined to be the polynomial of degree k and norm |Peer Reviewe