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Vladimir Nikiforov

Publications and source records attributed to Vladimir Nikiforov.

At least 55 records · Page 3Linked to original sources
arXiv2009

The spectral radius of graphs without paths and cycles of specified length

Let G be a graph with n vertices and mu(G) be the largest eigenvalue of the adjacency matrix of G. We study how large mu(G) can be when G does not contain cycles and paths of specified order. In particular, we determine the maximum spectral radius of graphs without paths of given length, and give tight bounds on the spectral radius of graphs without given even cycles. We also raise a number of natural open problems.

math.CO↗
arXiv2009

Degree powers in graphs with forbidden even cycle

We prove a conjecture of Yuster and Caro about the sum of the p-powers of the degrees of a graph of order n without a specified even cycle. Our proof is based on a new sufficient condition for long paths, that may be useful in other applications as well.

math.CO↗
arXiv2009

Spectral radius and Hamiltonicity of graphs

Let G be a graph of given order and mu(G) be the largest eigenvalue of its adjacency matrix. We give conditions on mu(G) that imply Hamiltonicity of G and of its complement.

math.CO↗
arXiv2008

The number of cliques in graphs of given order and size

Let k_r(n,m) denote the minimum number of r-cliques in graphs with n vertices and m edges. For r=3,4 we give a lower bound on k_r(n,m) that approximates k_r(n,m) with an error smaller than n^r/(n^2-2m). The solution is based on a constraint minimization of certain multilinear forms. In our proof, a combinatorial strategy is coupled with extensive analytical arguments.

math.CO↗
arXiv2007

Turan's theorem inverted

Turan's theorem implies that every graph of order n with more edges than the r-partite Turan graph contains a complete graph of order r+1. We show that the same premise implies the existence of much larger graphs. We also prove corresponding stability theorems. These results complete work started by Erdos in 1963.

math.CO↗
arXiv2007

A spectral stability theorem for large forbidden graphs

We extend the classical stability theorem of Erdos and Simonovits in two directions: first, we allow the order of the forbidden graph to grow as log of order of the host graph, and second, our extremal condition is on the spectral radius of the host graph.

math.CO↗
arXiv2007

Spectral saturation: inverting the spectral Turan theorem

We prove that if the spectral radius of a graph G of order n is larger than the spectral radius of the r-partite Turan graph of the same order, then G contains various supergraphs of the complete graph of order r+1. In particular G contains a complete r-partite graph of size log n with one edge added to the first part. These results complete a project of Erdos from 1963. We also give corresponding stability results.

math.CO↗
arXiv2007

Complete r-partite subgraphs of dense r-graphs

We determine how large r-partite graphs can be found in r-uniform graphs with n vertices and Cn^r edges, where C is a slowly decreasing function of n. This refines results of Erdos from 1964.

math.CO↗
arXiv2007

Ramsey Goodness and Beyond

In a seminal paper from 1983, Burr and Erdos 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 Szemeredi regularity lemma, embedding of sparse graphs, Turan type stability, and other structural results. We give exact Ramsey numbers for various classes of graphs, solving all but one of the Burr-Erdos problems.

math.CO↗
arXiv2007

Chromatic number and spectral radius

Let G be a graph, H be its chromatic number, L be the largest eigenvalue of its Laplacian, and M be the largest eigenvalue of its adjacency matrix. Then, complementing a well-known result of Hoffman, we show that L>=(H/(H-1))M

math.CO↗
arXiv2007

The spectral radius of subgraphs of regular graphs

We give a bound on the spectral radius of subgraphs of regular graphs with given order and diameter. We give a lower bound on the smallest eigenvalue of a nonbipartite regular graph of given order and diameter.

math.CO↗
arXiv2007

An Abstract Regularity Lemma

We extend Szemeredi's Regularity Lemma (SRL) to abstract measure spaces. Our main aim is to find general conditions under which the original proof of Szemeredi still works. To illustrate that our approach has some merit, we outline several applications. Some of these applications seem to be tailored to our approach: in particular, we are not aware of any alternative proofs.

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