Let G=(V,E) be a simple graph with n vertices and m edges, P(G,k) the chromatic polynomial of G, P(G,L) number L-colorings G for any k-assignment L. In this article, we show that when k≥m−1≥3, P(G,L)−P(G,k) is bounded below by ((k−m+1)kn−3+(k−m+3)c3kn−5)∑uv∈E|L(u)∖L(v)|, where c≥(m−1)(m−3)8, in particular, if K3-free, then c≥(m−22)+2m−3. Consequently, P(G,L)≥P(G,k) whenever k≥m−1.

Let k and n be integers such that k > 2n > 0. Let M be the complex analytic manifold defined by M = {x ∈ C : xx = 0, rank (x) = n}. Let G = SO(k,C) and G = GL(n,C), then Witt’s theorem on quadratic forms implies that G is a maximal connected Lie group acting transitively on M by right multiplication. Also, G is a maximal connected Lie group acting freely on M by left multiplication. If f ∈ C(M)...

LetG = (V,E) be a graph with a fixed orientation and let A be an abelian group. Let F (G,A) denote the set of all functions from E(G) to A. The graph G is A-connected if for any function f̄ ∈ F (G,A), there exists an A-flow f such that f(e) 6= f̄(e) for any e ∈ E(G). The graph G is A-colorable if for any function f ∈ F (G,A), there exists a function c : V (G) → A such that for any arc e = (u, v) ...

The notion of $k$-frames was recently introduced by Gu avruc ta in Hilbert  spaces to study atomic systems with respect to a bounded linear operator. A continuous frame is a family of vectors in a Hilbert space which allows reproductions of arbitrary elements by continuous super positions. In this manuscript, we construct a continuous $k$-frame, so called c$k$-frame along with an atomic system ...

For every positive integer k, a set S of vertices in a graph G = (V;E) is a k- tuple dominating set of G if every vertex of V -S is adjacent to at least k vertices and every vertex of S is adjacent to at least k - 1 vertices in S. The minimum cardinality of a k-tuple dominating set of G is the k-tuple domination number of G. When k = 1, a k-tuple domination number is the well-studied domination...

A Deza graph G with parameters (n,k,b,a) is a k-regular n vertices such that any two distinct have b or common neighbours, where b⩾a. The children GA and GB of are defined on the vertex set every adjacent in if only they respectively. In this paper we present general approach to dual Seidel switching investigate graphs whose strongly regular graphs.

Let k be a field, G an abelian group with a bicharacter, A a colour algebra; i.e., an associative graded k-algebra with identity, C a graded A-coring that is projective as a right A-module, C∗ the graded dual ring of C and Λ a left graded C-comodule that is finitely generated as a graded right C∗-module. We give necessary and sufficient conditions for projectivity and flatness of a graded modul...

Let k be an algebraically closed field of characteristic > 2, F = k((t)) and G = Sp 2d . In this paper we propose a geometric analog of the Weil representation of the metaplectic group G̃(F ). This is a category of perverse sheaves on some stack, on which G̃(F ) acts by functors. This construction will be used in [5] (and subsequent publications) for the proof of the geometric Langlands functoria...

We give an explicit combinatorial Chevalley-type formula for the equivariant K-theory of generalized flag varieties G/P. The formula implies a simple combinatorial model for the characters of the irreducible representations of G and, more generally, for the Demazure characters. The construction is given in terms of a certain R-matrix, that is, a collection of operators satisfying the Yang-Baxte...