In this paper we study some of the structural properties of the set of all minimal total dominating functions () of cycles and paths and introduce the idea of function reducible graphs and function separable graphs. It is proved that a function reducible graph is a function separable graph. We shall also see how the idea of function reducibility is used to study the structure of for some classes of graphs.
The purpose of this paper is to study W(2, 2) Lie conformal algebra, which has a free ℂ[∂]-basis L, M such that [...] [LλL]=(∂+2λ)L,[LλM]=(∂+2λ)M,[MλM]=0 . In this paper, we study conformal derivations, central extensions and conformal modules for this Lie conformal algebra. Also, we compute the cohomology of this Lie conformal algebra with coefficients in its modules. In particular, we determine its cohomology with trivial coefficients both for the basic and reduced complexes.
Let r ≥ 2 be an integer. A real number α ∈ [0,1) is a jump for r if for any ε > 0 and any integer m ≥ r, any r-uniform graph with n > n₀(ε,m) vertices and density at least α+ε contains a subgraph with m vertices and density at least α+c, where c = c(α) > 0 does not depend on ε and m. A result of Erdös, Stone and Simonovits implies that every α ∈ [0,1) is a jump for r = 2. Erdös asked whether the same is true for r ≥ 3. Frankl and Rödl gave a negative answer by showing an infinite sequence...
In this paper, we describe the sublattices of some lattices, extending previous results of [Ber]. Our description makes intensive use of graphs.
For a set of graphs, an -factor of a graph is a spanning subgraph of , where each component of is contained in . It is very interesting to investigate the existence of factors in a graph with given minimum degree from the prospective of eigenvalues. We first propose a tight sufficient condition in terms of the -spectral radius for a graph involving minimum degree to contain a star factor. Moreover, we also present tight sufficient conditions based on the -spectral radius and the distance...
A hypergraph is a sum hypergraph iff there are a finite S ⊆ IN⁺ and d̲, [d̅] ∈ IN⁺ with 1 < d̲ ≤ [d̅] such that is isomorphic to the hypergraph where V = S and . For an arbitrary hypergraph the sum number σ = σ() is defined to be the minimum number of isolated vertices such that is a sum hypergraph. Generalizing the graph Cₙ we obtain d-uniform hypergraphs where any d consecutive vertices of Cₙ form an edge. We determine sum numbers and investigate properties of sum labellings for this...