Here, we achieve a similar bound to above without requiring the graph to be random. The graph has a more relaxed assumption based on density and regularity. Definition 1 (Density and Regularity) For A,B ∈ V such that A ∩ B = ∅ and |A|, |B| > 1, let e(A,B) denote the number of edges between A and B, and let d(A,B) = e(A,B) |A||B| be the density. (A,B) is γ-regular if it has the following propert...

Let Γ denote a D-bounded distance-regular graph, where D ≥ 3 is the diameter of Γ. For 0 ≤ s ≤ D − 3 and a weak-geodetically closed subgraph ∆ of Γ with diameter s, define a graph G(∆) whose vertex set is the collection of all weak-geodetically closed subgraphs of diameter s+2 containing ∆, and vertex Ω is adjacent to vertex Ω′ in G if and only if Ω∩Ω′ as a subgraph of Γ has diameter s+1. We sh...

Let Γ be a group together with a sequence of normal subgroups Γ ⊃ Γ1 ⊃ Γ2 ⊃ . . . of finite index [Γ : Γk] such that T k Γk = {1}. Let (X,Y ) be a (compact) 4n-dimensional Poincaré pair and p : (X,Y )→ (X,Y ) be a Γ-covering, i.e. normal covering with Γ as deck transformation group. We get associated Γ/Γk-coverings (Xk, Yk)→ (X,Y ). We prove that sign(X,Y ) = lim k→∞ sign(Xk, Yk) [Γ : Γk] , whe...

An incompressible bounded surface F on the boundary of a compact, connected, orientable 3-manifold M is arc-extendible if there is a properly embedded arc γ on ∂M − IntF such that F ∪ N(γ) is incompressible, where N(γ) is a regular neighborhood of γ in ∂M . Suppose for simplicity that M is irreducible and F has no disk components. If M is a product F × I, or if ∂M −F is a set of annuli, then cl...

We combine the theory of Coxeter groups, the covering theory of graphs introduced by Malnic, Nedela and Skoviera and the theory of reflections of graphs in order to obtain the following characterization of a Coxeter group: Let π : Γ → (v,D, ι,−1) be a 1-covering of a monopole admitting semiedges only. The graph Γ is the Cayley graph of a Coxeter group if and only if π is regular and any deck tr...

Let Γ be an edge-regular graph with given parameters (v, k, λ). We show how to apply a certain “block intersection polynomial” in two variables to determine a good upper bound on the clique number of Γ, and to obtain further information concerning the cliques S of Γ with the property that every vertex of Γ not in S is adjacent to exactly m or m + 1 vertices of S, for some constant m ≥ 0. Some i...

We first show that every γ-contractive commuting multioperator is unitarily equivalent to the restriction of S⊕W to an invariant subspace, where S is a backwards multi-shift and W a γ-isometry. We then describe γ-isometries in terms of (γ, 1)-isometries, and establish that under an additional assumption on T , W above can be chosen to be a commuting multioperator of isometries. Our methods prov...

For 3 ≤ k ≤ 20 with k 6= 4, 8, 12, all the smallest currently known k–regular graphs of girth 5 have the same orders as the girth 5 graphs obtained by the following construction: take a (not necessarily Desarguesian) elliptic semiplane S of order n− 1 where n = k − r for some r ≥ 1; the Levi graph Γ (S) of S is an n–regular graph of girth 6; parallel classes of S induce co–cliques in Γ (S), som...

The word problem for inverse monoids generated by a set Γ subject to relations of the form e = f , where e and f are both idempotents in the free inverse monoid generated by Γ , is investigated. It is shown that for every fixed monoid of this form the word problem can be solved in polynomial time which solves an open problem of Margolis and Meakin. For the uniform word problem, where the presen...

This note completely resolves the asymptotic development of order 2 by Γ-convergence of the mass-constrained Cahn–Hilliard functional, by showing that one of the critical assumptions of the authors’ previous work [10] is unnecessary.