We give a simple proof based on symmetries that there are no geodesics from vertex to itself in the cube, tetrahedron, octahedron, and icosahedron.

We use the worldline representation of the one loop fermionic effective action to obtain the vertex functions that describe mesons in a possible string representation of the large N QCD. These vertex functions are then applied to study the chiral limit of the theory. We find that the propagator constructed from the vertex function V0, which would describes a tachyon in the open Nambu-Goto boson...

Let G = (V, E) be a simple graph on vertex set V and define a function f : V → {−1, 1}. The function f is a signed dominating function if for every vertex x ∈ V , the closed neighborhood of x contains more vertices with function value 1 than with −1. The signed domination number of G, γs(G), is the minimum weight of a signed dominating function on G. Let G denote the complement of G. In this pa...

It is known that a 3-edge-connected graph has a spanning even subgraph in which every component contains at lest five vertices, and the lower bound is best possible. A natural question arises whether we can improve the lower bound by changing the spanning property with the dominating property. In this paper, we show that a 3-edge-connected cubic graph has a dominating even subgraph in which eve...

Suppose that Γ is a connected graph and G is a subgroup of the automorphism group Aut(Γ) of Γ. Then Γ is G-symmetric if G acts transitively on the arcs (and so the vertices) of Γ and Γ is G-semisymmetric if G acts edge transitively but not vertex transitively on Γ. If Γ is Aut(Γ)symmetric, respectively, Aut(Γ)-semisymmetric, then we say that Γ is symmetric, respectively, semisymmetric. If Γ is ...

We generalize the embedding order to shuffle on trajectories and examine its properties. We give general characterizations of reflexivity, transitivity, anti-symmetry and other properties of binary relations, and investigate the associated decision problems. We also examine the property of convexity of a language with respect to these generalized embedding relations.

A k-container C(u, v) of G between u and v is a set of k internally disjoint paths between u and v. A k-container C(u, v) of G is a k∗-container if it contains all vertices of G. A graph G is k∗connected if there exists a k∗-container between any two distinct vertices. In this paper, we prove that the n-dimensional fully connected cubic network FCCNn is 3 ∗-connected for n ≥ 2.

An in nite family of cubic edge transitive but not vertex transitive graph graphs with edge stabilizer isomorphic to Z is constructed

Menger's Theorem states that in a 3-connected graph, any two vertices are joined by three openly disjoint paths. Here we consider 3-connected cubic graphs where two vertices exist so that the three disjoint paths between them contain all of the vertices of the graph (we call these graphs 3*connected); and also where the latter is true for ALL pairs of vertices (globally 3*-connected). A necessa...

While discrepancy theory is normally only studied in the context of 2-colorings, we explore the problem of k-coloring, for k 2, a set of vertices to minimize imbalance among a family of subsets of vertices. The imbalance is the maximum, over all subsets in the family, of the largest diierence between the size of any two color classes in that subset. The discrepancy is the minimum possible imbal...