Dagger compact closed categories were recently introduced by Abramsky and Coecke, under the name “strongly compact closed categories”, as an axiomatic framework for quantum mechanics. We present a graphical language for dagger compact closed categories, and sketch a proof of its completeness for equational reasoning. We give a general construction, the CPM construction, which associates to each...

An ear decomposition of a connected undirected graph G= (V ,E) is a partition of the edge (link) set E into a collection of edge-disjoint paths P0,P1, . . . ,P|E|−|V |+1, called ears, such that P0 is a link, P0 ∪P1 is a closed path (i.e., a cycle), and for each i , 2 i |E| − |V | + 1, Pi is a path of which each terminating node belongs to some Pj , j < i , and no internal node belongs to any Pj...

A vertex-magic total labeling on a graph G is a one-to-one map λ from V (G) ∪E(G) onto the integers 1, 2, · · · , |V (G) ∪E(G)| with the property that, given any vertex x, λ(x) + ∑ y∼x λ(y) = k for some constant k. In this paper we completely determine which complete bipartite graphs have vertex-magic total labelings.

In the present paper, we study some properties of fuzzy norm of linear operators. At first the bounded inverse theorem on fuzzy normed linear spaces is investigated. Then, we prove Hahn Banach theorem, uniform boundedness theorem and closed graph theorem on fuzzy normed linear spaces. Finally the set of all compact operators on these spaces is studied.

In this paper, the exact formulae for the generalized degree distance, degree distance and reciprocal degree distance of strong product of a connected and the complete multipartite graph with partite sets of sizes m0, m1, . . . , mr&minus1 are obtained. Using the results obtained here, the formulae for the degree distance and reciprocal degree distance of the closed and open fence graphs are co...

Graphs and digraphs treated here are finite and simple. Let G = (V (G), E(G)) be a connected graph with vertex set V (G) and edge set E(G), and D the symmetric digraph corresponding toG. SetD(G) = {(u, v), (v, u) | uv ∈ E(G)}. For e = (u, v) ∈ D(G), set u = o(e) and v = t(e). Furthermore, let e−1 = (v, u) be the inverse of e = (u, v). A path P of length n in G is a sequence P = (e1, · · · , en)...