Vladimir Nikiforov

Some Inequalities for the Largest Eigenvalue of a Graph

Let λ(G) be the largest eigenvalue of the adjacency matrix of a graph G: We show that if G is Kp+1-free then ***** insert CODING here *****This inequality was first conjectured by Edwards and Elphick in 1983 and supersedes a series of previous results on upper bounds of λ(G).Let Ti denote the number of all i-cliques of G, λ = λ(G) and p = cl(G): We show ***** insert equation here *****Let δ be the minimal degree of G. We show ***** insert equation here *****This inequality supersedes inequalities of Stanley and Hong. It is sharp for regular graphs and for a class of graphs which are in some sense maximally irregular.

The number of cliques in graphs of given order and size

Let kr (n, m) denote the minimum number of r-cliques in graphs with n vertices and m edges. We give a lower bound on kr (n, m) that approximates kr (n, m) with an error smaller than n r / n 2 − 2m � . This essentially solves a sixty year old problem. The solution is based on a constraint minimization of certain multilinear forms. In our proof, a combinatorial strategy is coupled with extensive analytical arguments.

Hermitian matrices and graphs: singular values and discrepancy

Abstract Let A =( a ij ) i , j =1 n be a Hermitian matrix of size n ⩾2, and set ρ(A)= 1 n 2 ∑ i,j=1 n a ij , disc (A)= max X,Y⊂[n],X≠∅,Y≠∅ 1 |X||Y| ∑ i∈X ∑ j∈Y (a ij −ρ(A)) . We show that the second singular value σ 2 ( A ) of A satisfies σ 2 (A)⩽C 1 disc (A) log n for some absolute constant C 1 , and this is best possible up to a multiplicative constant. Moreover, we construct infinitely many dense regular graphs G such that σ 2 (A(G))⩾C 2 disc (A(G)) log |G|, where C 2 >0 is an absolute constant and A ( G ) is the adjacency matrix of G . In particular, these graphs disprove two conjectures of Fan Chung.

Ramsey goodness and beyond

In a seminal paper from 1983, Burr and Erdős started the systematic study of Ramsey numbers of cliques vs. large sparse graphs, raising a number of problems. In this paper we develop a new approach to such Ramsey problems using a mix of the Szemerédi regularity lemma, embedding of sparse graphs, Turán type stability, and other structural results. We give exact Ramsey numbers for various classes of graphs, solving five — all but one — of the Burr-Erdős problems.

Graphs with many r-cliques have large complete r-partite subgraphs

Let r≥2 and c>0. Every graph on n vertices with at least cn r cliques on r vertices contains a complete r-partite subgraph with r−1 parts of size ⌊ c r log n⌋ and one part of size greater than n 1−cr−1 . This result implies a quantitative form of the Erdos-Stone theorem.

Note: The sum of the squares of degrees: Sharp asymptotics

Let f(n,m) be the maximum of the sum of the squares of degrees of a graph with n vertices and m edges. Summarizing earlier research, we present a concise, asymptotically sharp upper bound on f(n,m), better than the bound of de Caen for almost all n and m.

Cliques and the spectral radius

We prove a number of relations between the number of cliques of a graph G and the largest eigenvalue @m(G) of its adjacency matrix. In particular, writing ks(G) for the number of s-cliques of G, we show that, for all r>=2,@m^r^+^1(G)= =(@m(G)n-1+1r)r(r-1)r+1(nr)^r^+^1.

The Cycle-Complete Graph Ramsey Numbers

In 1978 Erdos, Faudree, Rousseau and Schelp conjectured that \[ r ( C_{p},K_{r} ) = ( p-1 ) (r-1) +1 \] for every

Spectral radius and Hamiltonicity of graphs with large minimum degree

p\,{\geq}\,r\,{\geq}\,3

Books in graphs

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