On graphs with an eigenvalue of maximal multiplicity

Let G be a graph of order n with an eigenvalue μ≠-1,0 of multiplicity k<n-2. It is known that k≤n+√2-2n+¼, equivalently k≤½t(t-1), where t=n-k>2. The only known examples with k=½t(t-1) are 3K2 (with n=6, μ=1, k=3) and the maximal exceptional graph G36 (with n=36, μ=-2, k=28). We show that no other example can be constructed from a strongly regular graph in the same way as G36 is constructed from the line graph L(K9)

Spectral preorder and perturbations of discrete weighted graphs

In this article, we introduce a geometric and a spectral preorder relation on the class of weighted graphs with a magnetic potential. The first preorder is expressed through the existence of a graph homomorphism respecting the magnetic potential and fulfilling certain inequalities for the weights. The second preorder refers to the spectrum of the associated Laplacian of the magnetic weighted graph. These relations give a quantitative control of the effect of elementary and composite perturbations of the graph (deleting edges, contracting vertices, etc.) on the spectrum of the corresponding Laplacians, generalising interlacing of eigenvalues. We give several applications of the preorders: we show how to classify graphs according to these preorders and we prove the stability of certain eigenvalues in graphs with a maximal d-clique. Moreover, we show the monotonicity of the eigenvalues when passing to spanning subgraphs and the monotonicity of magnetic Cheeger constants with respect to the geometric preorder. Finally, we prove a refined procedure to detect spectral gaps in the spectrum of an infinite covering graph.Comment: 26 pages; 8 figure

Notes on simplicial rook graphs

The simplicial rook graph ${\rm SR}(m,n)$ is the graph of which the vertices are the sequences of nonnegative integers of length $m$ summing to $n$, where two such sequences are adjacent when they differ in precisely two places. We show that ${\rm SR}(m,n)$ has integral eigenvalues, and smallest eigenvalue $s = \max (-n, -{m \choose 2})$, and that this graph has a large part of its spectrum in common with the Johnson graph $J(m+n-1,n)$. We determine the automorphism group and several other properties

Topologically biased random walk with application for community finding in networks

We present a new approach of topology biased random walks for undirected networks. We focus on a one parameter family of biases and by using a formal analogy with perturbation theory in quantum mechanics we investigate the features of biased random walks. This analogy is extended through the use of parametric equations of motion (PEM) to study the features of random walks {\em vs.} parameter values. Furthermore, we show an analysis of the spectral gap maximum associated to the value of the second eigenvalue of the transition matrix related to the relaxation rate to the stationary state. Applications of these studies allow {\em ad hoc} algorithms for the exploration of complex networks and their communities.Comment: 8 pages, 7 figure

Spectrum of Cayley graphs on the symmetric group generated by transpositions

For an integer $n\geq 2$, let $X_n$ be the Cayley graph on the symmetric group $S_n$ generated by the set of transpositions ${(1 2),(1 3),...,(1 n)}$. It is shown that the spectrum of $X_n$ contains all integers from $-(n-1)$ to $n-1$ (except 0 if $n=2$ or $n=3$).Comment: We have been informed by Guillaume Chapuy, Valentin F\'eray and Paul Renteln, that the problem in question can be solved by exploiting certain properties of the Jucys-Murphy elements, discovered by Jucys and independently by Flatto, Odlyzko and Wale