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 $g$ be a simple graph. an independent set is a set ofpairwise non-adjacent vertices. the number of vertices in a maximum independent set of $g$ isdenoted by $alpha(g)$. in this paper, we characterize graphs $g$ with $n$ vertices and with maximumnumber of maximum independent sets provided that $alpha(g)leq 2$ or $alpha(g)geq n-3$.

the vertex arboricity $rho(g)$ of a graph $g$ is the minimum number of subsets into which the vertex set $v(g)$ can be partitioned so that each subset induces an acyclic graph‎. ‎a graph $g$ is called list vertex $k$-arborable if for any set $l(v)$ of cardinality at least $k$ at each vertex $v$ of $g$‎, ‎one can choose a color for each $v$ from its list $l(v)$ so that the subgraph induced by ev...

In this paper, the notions of countable S∗-compactness is introduced in L-topological spaces based on the notion of S∗-compactness. An S∗-compact L-set is countably S∗-compact. If L = [0, 1], then countable strong compactness implies countable S∗-compactness and countable S∗-compactness implies countable F-compactness, but each inverse is not true. The intersection of a countably S∗-compact L-s...

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 L be a lattice with the least element 0. An element x ∈ L is a zero divisor if x∧ y = 0 for some y ∈ L∗ = L \ {0}. The set of all zero divisors is denoted by Z(L). We associate a simple graph Γ(L) to L with vertex set Z(L)∗ = Z(L) \ {0}, the set of non-zero zero divisors of L and distinct x, y ∈ Z(L)∗ are adjacent if and only if x ∧ y = 0. In this paper, we obtain certain properties and dia...

A frame $L$ is called {it coz-dense} if $Sigma_{coz(alpha)}=emptyset$ implies $alpha=mathbf 0$. Let $mathcal RL$ be the ring of real-valued continuous functions on a coz-dense and completely regular frame $L$. We present a description of the socle of the ring $mathcal RL$ based on minimal ideals of $mathcal RL$ and zero sets in pointfree topology. We show that socle of $mathcal RL$ is an essent...

When the iteration proceeds, the size l of labeled set L will grow larger and larger, so inverting the l × l matrix KL,L (KL,L is the sub-matrix of K corresponding to L) at scratch under each iteration is very inefficient. Here we tackle this computational issue by incrementally updating K−1 L,L based on the previous inverting result. As our submission, T is used to denote the curriculum set wi...

We introduce a hierarchy of problems between the Dominating Set problem and the Power Dominating Set (PDS) problem called the l-round power dominating set (l-round PDS, for short) problem. For l = 1, this is the Dominating Set problem, and for l ≥ n − 1, this is the PDS problem; here n denotes the number of nodes in the input graph. In PDS the goal is to find a minimum size set of nodes S that ...