Let G be a simple graph of order $$n\ge 2$$ and let $$k\in \{1,\ldots ,n-1\}$$ . The k-token $$F_k(G)$$ is the whose vertices are k-subsets V(G), where two adjacent in whenever their symmetric difference an edge G. In 2018 Leaños Trujillo-Negrete proved that if t-connected $$t\ge k$$ , then at least $$k(t-k+1)$$ -connected. this paper we show such lower bound remains true context edge-connectiv...

The l-component (edge) connectivity of a graph G, denoted by cκl(G) (cλl(G)), is the minimum number vertices (edges) whose removal from G results in disconnected with at least l components. pancake Pn popular underlying topology for distributed systems. In paper, we determine cκl(Pn) and cλl(Pn) 3 ≤ 5.

Topology control is one of the major approaches to achieve the energy efficiency as well as fault tolerance in wireless networks. In this paper, we study the dual power assignment problem for 2-edge connectivity and 2vertex connectivity in the symmetric graphical model. The problem is arisen from the following practical origin. In a wireless ad hoc network where each node can switch its transmi...

In this paper, we prove that the connectivity and the edge connectivity of the lexicographic product of two graphs G1 and G2 are equal to κ1v2 and min{λ1v 2 2 , δ2 +δ1v2}, respectively, where δi, κi, λi and vi denote the minimum degree, the connectivity, the edge-connectivity and the number of vertices of Gi, respectively. We also obtain that the edge-connectivity of the direct product of K2 an...

In this paper we discuss about tenacity and its properties in stability calculation. We indicate relationships between tenacity and connectivity, tenacity and binding number, tenacity and toughness. We also give good lower and upper bounds for tenacity. Since we are primarily interested in the case where disruption of the graph is caused by the removal of a vertex or vertices (and the resulting...

Short cycle connectivity is a generalization of ordinary connectiv-ity. Instead by a path (sequence of edges), two vertices have to be connected by a sequence of short cycles, in which two adjacent cycles have at least one common vertex. If all adjacent cycles in the sequence share at least one edge, we talk about edge short cycles connectivity. It is shown that the short cycles connectivity is...

The odd edge connectivity of a graph G, denoted by o(G), is the size of a smallest odd edge cut of the graph. Let S be any given surface and be a positive real number. We proved that there is a function fS( ) (depends on the surface S and lim !0 fS( )1⁄41) such that any graph G embedded in S with the odd-edge connectivity at least fS( ) admits a nowhere-zero circular (2þ )-flow. Another major r...