Spectrum of Markov generators on sparse random graphs

Correction in Proposition 4.3. Final version.International audienceWe investigate the spectrum of the infinitesimal generator of the continuous time random walk on a randomly weighted oriented graph. This is the non-Hermitian random nxn matrix L defined by L(j,k)=X(j,k) if kj and L(j,j)=-sum(L(j,k),kj), where X(j,k) are i.i.d. random weights. Under mild assumptions on the law of the weights, we establish convergence as n tends to infinity of the empirical spectral distribution of L after centering and rescaling. In particular, our assumptions include sparse random graphs such as the oriented Erdős-Rényi graph where each edge is present independently with probability p(n)->0 as long as np(n) >> (log(n))^6. The limiting distribution is characterized as an additive Gaussian deformation of the standard circular law. In free probability terms, this coincides with the Brown measure of the free sum of the circular element and a normal operator with Gaussian spectral measure. The density of the limiting distribution is analyzed using a subordination formula. Furthermore, we study the convergence of the invariant measure of L to the uniform distribution and establish estimates on the extremal eigenvalues of L

The Research on the L(2,1)-labeling problem from Graph theoretic and Graph Algorithmic Approaches

The L(2,1) -labeling problem has been extensively researched on many graph classes. In this thesis, we have also studied the problem on some particular classes of graphs. In Chapter 2 we present a new general approach to derive upper bounds for L(2,1)-labeling numbers and applied that approach to derive bounds for the four standard graph products. In Chapter 3 we study the L(2,1)-labeling number of the composition of n graphs. In Chapter 4 we consider the Cartesian sum of graphs and derive, both, lower and upper bounds for their L(2,1)-labeling number. We use two different approaches to derive the upper bounds and both approaches improve previously known bounds. We also present new approximation algorithms for the L(2,1 )-labeling problem on Cartesian sum graphs. In Chapter 5, we characterize d-disk graphs for d\u3e1, and give the first upper bounds on the L(2,1)-labeling number for this class of graphs. In Chapter 6, we compute upper bounds for the L(2,1)-labeling number of total graphs of K_{1,n}-free graphs. In Chapter 7, we study the four standard products of graphs using the adjacency matrix analysis approach. In Chapter 8, we determine the exact value for the L(2,1)-labeling number of a particular class of Mycielski graphs. We also provide, both, lower and upper bounds for the L(2,1)-labeling number of any Mycielski graph

Graphs with few 3-cliques and 3-anticliques are 3-universal

For given integers k, l we ask whether every large graph with a sufficiently small number of k-cliques and k-anticliques must contain an induced copy of every l-vertex graph. Here we prove this claim for k=l=3 with a sharp bound. A similar phenomenon is established as well for tournaments with k=l=4.Comment: 12 pages, 1 figur

On certain free product factors via an extended matrix model

Voiculescu's random matrix model for freeness is extended to the non-Gaussian case and also the case of constant block diagonal matrices. Thus we are able to investigate free products of free group factors with matrix algebras and with the hyperfinite II$_1$ factor, showing that $$L(F_n) * R = L(F_(n+1))$$ for $n \ge 1$, (where $L(F_1)=L(Z)$).Comment: 19+3 pages, AMSTeX 2.1 (figures in LaTeX 2.09), KD-at-UCB-00