We make the following three observations regarding a question popularized by Katznelson: is every subset of $${\mathbb {Z}}$$ which set Bohr recurrence also topological recurrence? (i) If G countable abelian group and $$E\subseteq G$$ an $$I_0$$ set, then $$E-E$$ recurrence. In particular $$\{2^n-2^m : n,m\in {\mathbb {N}}\}$$ (ii) Let {Z}}^{\omega }$$ be direct sum countably many copies with s...

In this paper we show upper bounds for the sum and the product of the lower domination parameters and the chromatic index of a graph. We also present some families of graphs for which these upper bounds are achieved. Next, we give a lower bound for the sum of the upper domination parameters and the chromatic index. This lower bound is a function of the number of vertices of a graph and a new gr...

the edge szeged index is a new molecular structure descriptor equal to the sum of products mu(e)mv(e) over all edges e = uv of the molecular graph g, where mu(e) is the number of edges which its distance to vertex u is smaller than the distance to vertex v, and nv(e) is defined analogously. in this paper, the edge szeged index of one-pentagonal carbon nanocone cnc5[n] is computed for the first ...

let $g$ be a simple connected graph. the edge-wiener index $w_e(g)$ is the sum of all distances between edges in $g$, whereas the hyper edge-wiener index $ww_e(g)$ is defined as {footnotesize $w{w_e}(g) = {frac{1}{2}}{w_e}(g) + {frac{1}{2}} {w_e^{2}}(g)$}, where {footnotesize $ {w_e^{2}}(g)=sumlimits_{left{ {f,g} right}subseteq e(g)} {d_e^2(f,g)}$}. in this paper, we present explicit formula fo...

Let G[H] be the lexicographic product of graphs G and H and let G⊕H be their Cartesian sum. It is proved that if G is a nonbipartite graph, then for any graph H, χ(G[H]) ≥ 2χ(H)+d k e, where 2k+1 is the length of a shortest odd cycle of G. Chromatic numbers of the Cartesian sum of graphs are also considered. It is shown in particular that for χ–critical and not complete graphs G and H, χ(G ⊕ H)...

The chromatic number of a graph G, denoted by χ(G), is the minimum number of colors such that G can be colored with these colors in such a way that no two adjacent vertices have the same color. A clique in a graph is a set of mutually adjacent vertices. The maximum size of a clique in a graph G is called the clique number of G. The Turán graph Tn(k) is a complete k-partite graph whose partition...

An outer-1-planar graph is a admitting drawing in the plane so that all vertices appear outer region of and every edge crosses at most one other edge. This paper establishes local structure graphs by proving each contains seventeen fixed configurations, list those configurations minimal sense for configuration there exist containing this do not contain any another sixteen configurations. There ...

The edge clique cover sum number (resp. edge clique partition sum number) of a graph G, denoted by scc(G) (resp. scp(G)), is defined as the smallest integer k for which there exists a collection of complete subgraphs of G, covering (resp. partitioning) all edges of G such that the sum of sizes of the cliques is at most k. By definition, scc(G) 5 scp(G). Also, it is known that for every graph G ...

An injective function f : V pGq Ñ t0, 1, 2, . . . , qu is an odd sum labeling if the induced edge labeling f defined by f puvq fpuq fpvq, for all uv P EpGq, is bijective and f pEpGqq t1, 3, 5, . . . , 2q 1u. A graph is said to be an odd sum graph if it admits an odd sum labeling. In this paper we study the odd sum property of graphs obtained by duplicating any edge of some graphs.