A generalization of Turán's theorem (original) (raw)

On the order of regular graphs with fixed second largest eigenvalue

arXiv (Cornell University), 2018

Let v(k, λ) be the maximum number of vertices of a connected k-regular graph with second largest eigenvalue at most λ. The Alon-Boppana Theorem implies that v(k, λ) is finite when k > λ 2 +4 4 . In this paper, we show that for fixed λ ≥ 1, there exists a constant C(λ) 4 . * J.Y.

Regular graphs with small second largest eigenvalue

Applicable Analysis and Discrete Mathematics, 2013

We consider regular graphs with small second largest eigenvalue (denoted by ?2). In particular, we determine all triangle-free regular graphs with ?2 ? ?2, all bipartite regular graphs with ?2 ? ?3, and all bipartite regular graphs of degree 3 with ?2 ? 2.

A generalization of Turán's theorem to directed graphs

Discrete Mathematics, 1980

We consider An extremal problem for directed graphs which is closely related to Tutin's theorem giving the maximum number of edges in a gr;lph on n vertices which does not contain a complete subgraph on m vertices. For an ;ntc&r n 22, let T,, denote the transitive tournament with vertex set X,, = {1,2,3,. .. , n) and edge set {(i. j): 1 s i C j s n]. A subgraph H of T,, is said to be m-locally unipathic when the restriction of H to each m element subset of X,, consisting of m consecutive integers is unipathic. We show that the maximum number of edges in a m-locally unipathic subgraph of T,, is (;g)(m-l)'+q(m-1)r + Ur"] where n = q(m-1) + r and [$<m-I)] s rc' @rn-1)1. As is the case with T&n's theorem, the extremal graphs for our problem are complete multipartite graphs. Unlike T:r&n's theorem, the part sizes will not be uniform The proof of our principal theorem rests 011 a combiaatorial theory originally developed to inves:dgate the rank of partial'iy ordered sets. For integers, n, k with n se k a 2, let g(n, k) be the maximum number of edges in a graph G on n vertices which does not contain a complete subgraph on k vertices. Then let n = (k-1)q + r where 0 6 r C k-1 and consider the complete multipartite graph G(n, k) having k-1r parts of size q and r parts of size q f 1. Clearly, G(n, k) has n vertices but does not have a complete subgraph on k vertices. The following well known theorem of P'. Tur6n [9] tell us that the lower bound on g(n, k) provided by the graph G(n, k) is best possible. It also Sells us that G(n, k) is the unique extremal graph. Theorem 1 (Turhn). For integers m, k with n > k 2~ 2 the maximurn number g(n, k) of edges in a graph on n vertices which does not contain a complete subgraph on k

An Extremal Property of Turán Graphs

Electronic Journal of Combinatorics, 2010

Let F n,tr(n) denote the family of all graphs on n vertices and t r (n) edges, where t r (n) is the number of edges in the Turán's graph T r (n)-the complete r-partite graph on n vertices with partition sizes as equal as possible. For a graph G and a positive integer λ, let P G (λ) denote the number of proper vertex colorings of G with at most λ colors, and let f (n, t r (n), λ) = max{P G (λ) : G ∈ F n,tr(n) }. We prove that for all n ≥ r ≥ 2, f (n, t r (n), r + 1) = P Tr(n) (r + 1) and that T r (n) is the only extremal graph.

A combinatorial bound on the number of distinct eigenvalues of a graph

2022

The smallest possible number of distinct eigenvalues of a graph G, denoted by q(G), has a combinatorial bound in terms of unique shortest paths in the graph. In particular, q(G) is bounded below by k, where k is the number of vertices of a unique shortest path joining any pair of vertices in G. Thus, if n is the number of vertices of G, then n − q(G) is bounded above by the size of the complement (with respect to the vertex set of G) of the vertex set of the longest unique shortest path joining any pair of vertices of G. The purpose of this paper is to commence the study of the minor-monotone floor of n − k, which is the minimum of n − k among all graphs of which G is a minor. Accordingly, we prove some results about this minor-monotone floor.

An Extremal Property of Turán Graphs, II

Journal of Graph Theory, 2013

Let T 2 (n) denote Turán's graph-the complete 2-partite graph on n vertices with partition sizes as equal as possible. We show that for all n ≥ 4, the graph T 2 (n) has more proper vertex colorings in at most 4 colors than any other graph with the same number of vertices and edges. C 2013

Another extremal problem for Turan graphs

Discrete Mathematics, 1987

We consider only finite, undirected graphs without loops or multiple edges. A clique of a graph G is a maximal complete subgraph of G. The clique number w(G) is the number of vertices in the largest clique of G. This note addresses the foflowing question: Which graphs G on n vertices with w(G) = r have the maximum number of cliques?

Spectral upper bounds for the order of a k-regular induced subgraph

Linear Algebra and its Applications, 2010

Let G be a simple graph with least eigenvalue λ, and let S be a set of vertices in G which induce a subgraph with mean degree k. We use a quadratic programming technique in conjunction with the main angles of G to establish an upper bound of the form |S| ≤ inf{(k + t)q G (t) : t > −λ}, where q G is a rational function determined by the spectra of G and its complement. In the case k = 0 we obtain improved bounds for the independence number of various benchmark graphs.

A generalization of a Turán’s theorem about maximum clique on graphs

Electronic Journal of Graph Theory and Applications

One of the most important Turán's theorems establishes an inequality between the maximum clique and the number of edges of a graph. Since 1941, this result has received much attention and many of the different proofs involve induction and a probability distribution. In this paper we detail finite procedures that gives a proof for the Turán's Theorem. Among other things, we give a generalization of this result. Also we apply this results to a Nikiforov's inequality between the spectral radius and the maximum clique of a graph.

Edge-Connectivity, Eigenvalues, and Matchings in Regular Graphs

SIAM Journal on Discrete Mathematics, 2010

In this paper, we study the relationship between eigenvalues and the existence of certain subgraphs in regular graphs. We give a condition on an appropriate eigenvalue that guarantees a lower bound for the matching number of a t-edge-connected d-regular graph, when t ≤ d − 2. This work extends some classical results of von Baebler and Berge and more recent work of Cioabȃ, Gregory, and Haemers. We also study the relationships between the eigenvalues of a d-regular t-edge-connected graph G and the maximum number of pairwise disjoint connected subgraphs in G that are each joined to the rest of the graph by exactly t edges.