Oppositeness graphs of a spherical building are generalizations of the classical Kneser graphs. Recently Brouwer [1] has shown that the square of each eigenvalue of the adjacency matrix of an oppositeness graph is a power of q, for buildings of finite groups of Lie type defined over Fq. Here we show that the incidence modules are simple. The essential part of the proof is a result of Carter and...

The main result is a common generalization of results on lower bounds for the chromatic number of r-uniform hypergraphs and some of the major theorems in Tverbergtype theory, which is concerned with the intersection pattern of faces in a simplicial complex when continuously mapped to Euclidean space. As an application we get a simple proof of a generalization of a result of Kriz for certain par...

Lovazs’ proof of the Kneser Conjecture presents a beautiful application of Borsuk-Ulam Theorem, a purely topological result, to a combinatorial problem on finite graphs. Moreover, Kneser-Lovasz Theorem is far from being the only case in which a combinatorial problem admits as a solution that uses topological techniques. It is rather surprising that the study of continuous maps would yield elega...

A set system is called t-intersecting if every two members meet each other in at least t elements. Katona determined the minimum ratio of the shadow and the size of such families and showed that the Erdős– Ko–Rado theorem immediately follows from this result. The aim of this note is to reproduce the proof to obtain a slight improvement in the Kneser graph. We also give a brief overview of corre...

The vertex set of the reduced Kneser graph KG2(m, 2) consists of all pairs {a, b} such that a, b ∈ {1, 2, . . . ,m} and 2 ≤ |a−b| ≤ m−2. Two vertices are defined to be adjacent if they are disjoint. We prove that, if m ≥ 4 and m 6= 5, then the circular chromatic number of KG2(m, 2) is equal to m − 2, its ordinary chromatic number.

In 1978, Alexander Schrijver defined the stable Kneser graphs as a vertex critical subgraphs of graphs. early 2000s, Günter M. Ziegler generalized Schrijver’s construction and s-stable Thereafter Frédéric Meunier determined chromatic number for special cases formulated conjecture on this paper we study generalization For some specific values parameter show that neighborhood complex < s, t &g...

The toughness t ( G ) of a graph is measure its connectivity that closely related to Hamiltonicity. Xiaofeng Gu, confirming longstanding conjecture Brouwer, recently proved the lower bound ≥ ℓ / λ − 1 on any connected -regular graph, where largest nontrivial absolute eigenvalue adjacency matrix. Brouwer had also observed many families graphs (in particular, those achieving equality in Hoffman r...

Coherence is established by semantic connections between sentences of a text which can be modeled by lexical relations. In this paper, we introduce the lexical coherence graph (LCG), a new graph-based model to represent lexical relations among sentences. The frequency of subgraphs (coherence patterns) of this graph captures the connectivity style of sentence nodes in this graph. The coherence o...