We show that every plane graph of diameter 2r in which all inner faces are triangles and all inner vertices have degree larger than 5 can be covered with two balls of radius r : 2003 Wiley Periodicals, Inc. J Graph Theory 44:

The degree distance of a graph which is a degree analogue of the Wiener index of the graph. Let G = TUV C6[2p, q] be the carbon nanotubes covered by C6, formulas for calculating the degree distance of armchair polyhex nanotubes TUV C6[2p, q] are provided. Mathematics Subject Classification: 05C05, 05C12

Let Γ be a G-symmetric graph whose vertex set admits a nontrivial G-invariant partition B with block size v. Let ΓB be the quotient graph of Γ relative to B and Γ [B,C] the bipartite subgraph of Γ induced by adjacent blocks B,C of B. In this paper we study such graphs for which ΓB is connected, (G,2)-arc transitive and is almost covered by Γ in the sense that Γ [B,C] is a matching of v−1≥2 edge...

We show that for any 2-local colouring of the edges of a complete bipartite graph, its vertices can be covered with at most 3 disjoint monochromatic paths. And, we can cover almost all vertices of any complete or complete bipartite r-locally coloured graph with O(r) disjoint monochromatic cycles. MSC: 05C38, 05C55.

The following is proved: if every bridgeless graph G has a cycle cover of length at most 7/51€(G)I, then every bridgeless graph G has a cycle cover of length at most 7/51€(G)I such that any edge of G is covered once or twice.

‎let $r$ be a commutative ring with non-zero identity. ‎we describe all $c_3$‎- ‎and $c_4$-free intersection graph of non-trivial ideals of $r$ as well as $c_n$-free intersection graph when $r$ is a reduced ring. ‎also, ‎we shall describe all complete, ‎regular and $n$-claw-free intersection graphs. ‎finally, ‎we shall prove that almost all artin rings $r$ have hamiltonian intersection graphs. ...

The non-commuting graph $Gamma(G)$ of a non-abelian group $G$ with the center $Z(G)$ is a graph with thevertex set $V(Gamma(G))=Gsetminus Z(G)$ and two distinct vertices $x$ and $y$ are adjacent in $Gamma(G)$if and only if $xy neq yx$. The aim of this paper is to compute the spectra of some well-known NC-graphs.

Among the techniques for capturing nearby groundwater, covered horizontal well (draining gallery) stands out in its different types of water mine, qanat, and cimbre. The collected by these means is used to supply people livestock, irrigation, movement hydraulic devices, etc. Because they are carried surface gravity (without need energy) because only recharging groundwater table that takes place...

let $r$ be a commutative ring with identity and $m$ an $r$-module. in this paper, we associate a graph to $m$, say ${gamma}({}_{r}m)$, such that when $m=r$, ${gamma}({}_{r}m)$ coincide with the zero-divisor graph of $r$. many well-known results by d.f. anderson and p.s. livingston have been generalized for ${gamma}({}_{r}m)$. we show that ${gamma}({}_{r}m)$ is connected with ${diam}({gamma}({}_...