Simple-homotopy for simplicial and CW complexes is a special kind of topological homotopy constructed by elementary collapses and expansions. In this paper we introduce graph homotopy for graphs andâ€¦ (More)

The Euler characteristic plays an important role in many subjects of discrete and continuous mathematics. For noncompact spaces, its homological definition, being a homotopy invariant, seems not asâ€¦ (More)

This paper is to introduce circuit, bond, flow, and tension spaces and lattices for signed graphs, and to study the relations among these spaces and lattices. The key ingredient is to introduceâ€¦ (More)

Let Ïƒ be a simplex of RN with vertices in the integral lattice ZN . The number of lattice points of mÏƒ (= {mÎ± : Î± âˆˆ Ïƒ}) is a polynomial function L(Ïƒ,m) of m â‰¥ 0. In this paper we present: (i) aâ€¦ (More)

Graphene is an atomic thin two-dimensional semimetal whereas ZnO is a direct wide band gap semiconductor with a strong light-emitting ability. In this paper, we report on photoluminescence (PL) ofâ€¦ (More)

Gioan showed that the number of cycle reversing classes of totally cyclic orientations of a given graph can be calculated as an evaluation of the corresponding Tutte polynomial. We note that theâ€¦ (More)

Throughout this paper, we denote by V a finite dimensional vector space over an ordered field F with an inner product ( , ) . The indicator function of a subset E of V is the characteristic functionâ€¦ (More)