Let G = (V, E) be a digraph (i.e., a directed graph) with n vertices and m edges, and w : E → IR be a weight function on the edges. A directed cycle is closed walkC = (v0, v1, . . . , vt), where vt = v0 and (vi → vi+1) ∈ E, for i = 0, . . . , t − 1. The average cost of a directed cycle is AvgCost(C) = ω(C)/t = (∑e∈C ω(e)) /t. For each k = 0, 1, . . ., and v ∈ V , let dk(v) denote the minimum le...

Let G(V,E) be a graph with p vertices and q edges. A graph G is said to have an odd mean labeling if there exists a function f : V (G) → {0, 1, 2,...,2q - 1} satisfying f is 1 - 1 and the induced map f* : E(G) → {1, 3, 5,...,2q - 1} defined by f*(uv) = (f(u) + f(v))/2 if f(u) + f(v) is evenf*(uv) = (f(u) + f(v) + 1)/2 if f(u) + f(v) is odd is a bijection. A graph that admits an odd mean labelin...

Let M = (V, E, A) be a mixed graph with vertex set V , edge set E and arc set A. A cycle cover of M is a family C = {C1, . . . ,Ck} of cycles of M such that each edge/arc of M belongs to at least one cycle in C. The weight of C is ∑k i=1 |Ci |. The minimum cycle cover problem is the following: given a strongly connected mixed graph M without bridges, find a cycle cover of M with weight as small...

The threshold dimension of a graph G(V,E) is the smallest integer k such that E can be covered by k threshold spanning subgraphs of G. A k-dimensional box is the Cartesian product R1 ×R2 × · · · × Rk where each Ri is a closed interval on the real line. The boxicity of a graph G, denoted as box(G), is the minimum integer k such that G can be represented as the intersection graph of a collection ...

In this paper, we compare the computability and complexity of a continuous real function F with the computability and complexity of the graph G of the function F . A similar analysis will be carried out for functions on subspaces of the real line such as the Cantor space, the Baire space and the unit interval. In particular, we de1ne four basic types of e"ectively closed sets C depending on whe...

The perturbation semigroup was first defined in the case of $*$-algebras by Chamseddine, Connes and van Suijlekom. In this paper, we take $\mathcal{E}$ as a concrete operator system with unit. We give definition gauge group $\mathcal{G}(\mathcal{E})$ $\mathcal{E}$, after that closed respect to Haagerup tensor norm. also show there is continuous homomorphism from collection unital completely bou...

The (p)-neighborhood graph of a graph G, denoted N p (G), is de-ned on the same vertex set as G, with x; y] 2 E (N 2 (G)) if and only if jN(x)\N(y)j p in G, where N (v) is the open neighborhood of vertex v. The p]-neighborhood graph of G, N p G], is deened similarly, using closed neighborhoods rather than open ones. If G is the underlying graph of a symmetric digraph D, then the p-neighborhood ...

Let G be a subgraph-closed graph class with bounded maximum degree. We show that if G has balanced separators whose size is smaller than linear by a polynomial factor, then G has subexponential expansion. This gives a partial converse to a result of Nešetřil and Ossona de Mendez. As an intermediate step, the proof uses a new kind of graph decompositions. The concept of graph classes with bounde...

A new graph parameter, edge neighbor toughness is introduced to measure how difficult it for a be broken into many components through the deletion of closed neighborhoods few edges. Let G=(V,E) graph. An e said subverted when its neighborhood and two endvertices are deleted from G. set S⊆E(G) called an cut-strategy if all edges in S has been G survival subgraph, denoted by G/S, disconnected, or...