many graph theoretical parameters have been used to describe the vulnerability of communication networks, including toughness, binding number, rate of disruption, neighbor-connectivity, integrity, mean integrity, edgeconnectivity vector, l-connectivity and tenacity. in this paper we discuss integrity and its properties in vulnerability calculation. the integrity of a graph g, i(g), is defined t...

Let CMSO denote the counting monadic second order logic of graphs. We give a constructive proof that for some computable function f , there is an algorithm A that takes as input a CMSO sentence φ, a positive integer t, and a connected graph G of maximum degree at most ∆, and determines, in time f(|φ|, t) · 2O(∆·t) · |G|O(t), whether G has a supergraph G′ of treewidth at most t such that G′ |= φ...

‎let $d_{n,m}=big[frac{2n+1-sqrt{17+8(m-n)}}{2}big]$ and‎ ‎$e_{n,m}$ be the graph obtained from a path‎ ‎$p_{d_{n,m}+1}=v_0v_1 cdots v_{d_{n,m}}$ by joining each vertex of‎ ‎$k_{n-d_{n,m}-1}$ to $v_{d_{n,m}}$ and $v_{d_{n,m}-1}$‎, ‎and by‎ ‎joining $m-n+1-{n-d_{n,m}choose 2}$ vertices of $k_{n-d_{n,m}-1}$‎ ‎to $v_{d_{n,m}-2}$‎. ‎zhang‎, ‎liu and zhou [on the maximal eccentric‎ ‎connectivity ind...

Relative centricity RC values of vertices/atoms are calculated within the Distance Detour and Cluj-Distance criteria on their corresponding Shell transforms. The vertex RC distribution in a molecular graph gives atom equivalence classes, useful in interpretation of NMR spectra. Timed by vertex valences, RC provides a new index, called Centric Connectivity CC, which can be useful in the topologi...

Let d(u, v) denote the distance between two distinct vertices of a connected graph G, and diam(G) be the diameter of G. A radio labeling c of G is an assignment of positive integers to the vertices of G satisfying d(u, v) + |c(u) − c(v)| ≥ diam(G) + 1. The maximum integer in the range of the labeling is its span. The radio number of G, rn(G), is the minimum possible span. The family of gear gra...

In this paper, we discuss how to obtain certificates for the non-feasibility of connectivity graphs arising from multi-agent formations. We summarize some previous work on connectivity graphs and their feasibility. Next, we introduce the Positivstellensatz to show how it can be used to better understand the space of all connectivity graphs for a fixed number of vertices, which had previously be...

In this paper, the concept of distance local connectivity of a graph is presented. Several results involving the independence number of graphs and degree conditions for distance local connectivity of graphs are proved. The relations between distance local connectivity of graphs and the hamiltonian index of graphs are given.

Day and Tripathi [K. Day, A. Tripathi, Unidirectional star graphs, Inform. Process. Lett. 45 (1993) 123–129] proposed an assignment of directions on the star graphs and derived attractive properties for the resulting directed graphs. Cheng and Lipman [E. Cheng, M.J. Lipman, On the Day–Tripathi orientation of the star graphs: Connectivity, Inform. Process. Lett. 73 (2000) 5–10; E. Cheng, M.J. Li...

Prime graph of a ring R is a graph whose vertex set is the whole set R any any two elements $x$ and $y$ of $R$ are adjacent in the graph if and only if $xRy = 0$ or $yRx = 0$.  Prime graph of a ring is denoted by $PG(R)$. Directed prime graphs for non-commutative rings and connectivity in the graph are studied in the present paper. The diameter and girth of this graph are also studied in the pa...