Eigenvalues and forbidden subgraphs I
نویسنده
چکیده
Suppose a graph G have n vertices, m edges, and t triangles. Letting λn (G) be the largest eigenvalue of the Laplacian of G and μn (G) be the smallest eigenvalue of its adjacency matrix, we prove that λn (G) ≥ 2m2 − 3nt m (n2 − 2m) , μn (G) ≤ 3n3t− 4m3 nm (n2 − 2m) , with equality if and only if G is a regular complete multipartite graph. Moreover, if G is Kr+1-free, then λn (G) ≥ 2mn (r − 1) (n2 − 2m) with equality if and only if G is a regular complete r-partite graph.
منابع مشابه
Signless Laplacian eigenvalues and circumference of graphs
In this paper, we investigate the relation between the Q -spectrum and the structure of G in terms of the circumference of G. Exploiting this relation, we give a novel necessary condition for a graph to be Hamiltonian by means of its Q -spectrum. We also determine the graphs with exactly one or two Q -eigenvalues greater than or equal to 2 and obtain all minimal forbidden subgraphs and maximal ...
متن کاملMatrix partitions of perfect graphs
Given a symmetric m by m matrix M over 0, 1, ∗, the M -partition problem asks whether or not an input graph G can be partitioned into m parts corresponding to the rows (and columns) of M so that two distinct vertices from parts i and j (possibly with i = j) are nonadjacent if M(i, j) = 0, and adjacent if M(i, j) = 1. These matrix partition problems generalize graph colourings and homomorphisms,...
متن کاملThe Minimum Rank Problem over Finite Fields
Let Gk(F ) = {G | mr(F,G) ≤ k}, the set of simple graphs with minimum rank at most k. The problem of finding mr(F,G) and describing Gk(F ) has recently attracted considerable attention, particularly for the case in which F = R (see [Nyl96, CdV98, JD99, Hsi01, JS02, CHLW03, vdH03, BFH04, BvdHL04, HLR04, AHK05, BD05, BFH05a, BFH05b, BvdHL05, DK06, BF07]). The minimum rank problem over R is a sub-...
متن کاملA Multipartite Version of the Turan Problem - Density Conditions and Eigenvalues
In this paper we propose a multipartite version of the classical Turán problem of determining the minimum number of edges needed for an arbitrary graph to contain a given subgraph. As it turns out, here the non-trivial problem is the determination of the minimal edge density between two classes that implies the existence of a given subgraph. We determine the critical edge density for trees and ...
متن کاملForbidden Subgraphs for Planar Blict and Blitact Graphs
Many graphs which are encountered in the study of graph theory are characterized by a type of configuration or subgraphs they possess. However, there are occasions when such graphs are more easily defined or described by the kind of subgraphs they are not permitted to contain. Such subgraphs are called forbidden subgraphs. In this paper, we present characterizations of graphs whose blict and bl...
متن کامل