For n ∈ N and D ⊆ N, the distance graph P n has vertex set {0, 1, . . . , n− 1} and edge set {ij | 0 ≤ i, j ≤ n − 1, |j − i| ∈ D}. Note that the important and very well-studied circulant graphs coincide with the regular distance graphs. A fundamental result concerning circulant graphs is that for these graphs, a simple greatest common divisor condition, their connectivity, and the existence of ...

Given a smooth complex surface S, and a compact connected global normal crossings divisor D = ∪ i D i , we consider the local fundamental group π 1 (T \ D), where T is a good tubular neighbourhood of D. One has an exact sequence 1 → K → Γ := π 1 (T −D) → Π := π 1 (D) → 1, and the kernel K is normally generated by geometric loops γ i around the curve D i. Among the main results, which are strong...

Let G(V, E) be a graph with pvertices and qedges. f∶ V (G) → {0,1,2,…, p-1} bijection such that the induced function f*: E(G) N defined by f_sqdp* (uv)=|[f(u) ]2-[f(v) ]2 | for everyuv∈E(G).If is injective, then f_sqdp*is calledsquare difference labeling of G.A Gwhich admits square called graph. The greatest common incidence number (gcin) vertex v degree > 1 as divisor (g.c.d) labels inciden...

We generalize the computation of Feynman integrals of log divergent graphs in terms of the Kirchhoff polynomial to the case of graphs with both fermionic and bosonic edges, to which we assign a set of ordinary and Grassmann variables. This procedure gives a computation of the Feynman integrals in terms of a period on a supermanifold, for graphs admitting a basis of the first homology satisfying...

We give necessary and sufficient conditions under which the Jacobian of a graph is generated by a divisor that is the difference of two vertices. This answers a question posed by Becker and Glass and allows us to prove various other propositions about the order of divisors that are the difference of two vertices. We conclude with some conjectures about these divisors on random graphs and suppor...

In this article, we determine up to isomorphism of rings, rings R such that R has the following properties: (i) R is a commutative ring with identity which admits at least two nonzero zero-divisors, (ii) the complement of the zero-divisor graph of R is connected and it admits a cut vertex. Indeed, it is proved that there are exactly two such rings up to isomorphism of rings.

Let $I$ be a proper ideal of a commutative semiring $R$ and let $P(I)$ be the set of all elements of $R$ that are not prime to $I$. In this paper, we investigate the total graph of $R$ with respect to $I$, denoted by $T(Gamma_{I} (R))$. It is the (undirected) graph with elements of $R$ as vertices, and for distinct $x, y in R$, the vertices $x$ and $y$ are adjacent if and only if $x + y in P(I)...

The main purpose of this paper is to prove an important generalization of the construction of the Incidence Divisor given in [BMg1] in the case of an ambient manifold. Let us first recall briefly the setting : let Z be a complex manifold and (X s) s∈S an analytic family of (closed) n−cycles in Z parametrized by a reduced complex space S. To a (n+1)−codimensional subspace Y in Z, which is assume...

Let £ be a $0$-distributive lattice with the least element $0$, the greatest element $1$, and ${rm Z}(£)$ its set of zero-divisors. In this paper, we introduce the total graph of £, denoted by ${rm T}(G (£))$. It is the graph with all elements of £ as vertices, and for distinct $x, y in £$, the vertices $x$ and $y$ are adjacent if and only if $x vee y in {rm Z}(£)$. The basic properties of the ...

In 2008, J. Skowronek-kazi o ´ w extended the study of clique number id="M3"> ω G Z n to zero-divisor graph ring id="M4"> , but their result was imperfect. this ...