The spectral radius of a graph is the largest eigenvalue of its adjacency matrix. Let F (λ) be the family of connected graphs of spectral radius ≤ λ. We show that F (λ) can be defined by a finite set of forbidden subgraphs if and only if λ < λ * :The study of forbidden subgraphs characterization for F (λ) is motivated by the problem of estimating the maximum cardinality of equiangular lines in the n-dimensional Euclidean space R n -a family of lines through the origin such that the angle between any pair of them is the same. Denote by N α (n) the maximum number of equiangular lines in R n with angle arccos α. We establish the asymptotic formula N α (n) = c α n + O α (1) for every α ≥ 1 1+2λ * . In particular, N 1/3 (n) = 2n + O(1) and N 1/5 (n), N 1/(1+2 √ 2) (n) = 3 2 n + O(1). Besides we show that N α (n) ≤ 1.49n + O α (1) for every α = 1 3 , 1 5 , 1 1+2 √ 2 , which improves a recent result of

We show that, for each fixed k, an n-vertex graph not containing a cycle of length 2k has at most 80

A set of vertices S resolves a graph if every vertex is uniquely determined by its vector of distances to the vertices in S. The metric dimension of a graph is the minimum cardinality of a resolving set of the graph. Fix a connected graph G on q ≥ 2 vertices, and let M be the distance matrix of G. We prove that if there exists w ∈ Z q such that i w i = 0 and the vector M w, after sorting its coordinates, is an arithmetic progression with nonzero common difference, then the metric dimension of the Cartesian product of n copies of G is (2+o(1))n/ log q n. In the special case that G is a complete graph, our results close the gap between the lower bound attributed to Erdős and Rényi and the upper bounds developed subsequently

No abstract

We prove that any family E 1 , . . . , E ⌈rn⌉ of (not necessarily distinct) sets of edges in an r-uniform hypergraph, each having a fractional matching of size n, has a rainbow fractional matching of size n (that is, a set of edges from distinct E i 's which supports such a fractional matching). When the hypergraph is r-partite and n is an integer, the number of sets needed goes down from rn to rn − r + 1. The problem solved here is a fractional version of the corresponding problem about rainbow matchings, which was solved by Drisko and by Aharoni and Berger in the case of bipartite graphs, but is open for general graphs as well as for r-partite hypergraphs with r > 2. Our topological proof is based on a result of Kalai and Meshulam about a simplicial complex and a matroid on the same vertex set.