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 $alpha(g)geq n-3$.

In this paper we initialize the study of independent domination in directed graphs. We show that an independent dominating set of an orientation of a graph is also an independent dominating set of the underlying graph, but that the converse is not true in general. We then prove existence and uniqueness theorems for several classes of digraphs including orientations of complete graphs, paths, tr...

in this note we are going to survey several invariants of finite groups related either to their orders or to generating sets or to lattices of subgroups. some relations among these invariants will be exhibited. special attention will be paid to monotonicity of them.

A cactus graph is a connected graph in which every block is either an edge or a cycle. An optimal algorithm is presented here to find a maximum independent set and maximum 2-independent set on cactus graphs in O(n) time, where n is the total number of vertices of the graph. The cactus graph has many applications in real life problems, specially in radio communication system.

let $g=(v,e)$ be a simple graph. a set $ssubseteq v$ isindependent set of $g$,  if no two vertices of $s$ are adjacent.the  independence number $alpha(g)$ is the size of a maximumindependent set in the graph. in this paper we study and characterize the independent sets ofthe zero-divisor graph $gamma(r)$ and ideal-based zero-divisor graph $gamma_i(r)$of a commutative ring $r$.

let $r$ be a commutative ring. in this paper, by using algebraic properties of $r$, we study the hase digraph of prime ideals of $r$.

(Note that for circuits we implicitly assume that ∅ / ∈ C(M), just as we assume that for matroids ∅ ∈ I.) Proof: 1. follows from the definition that a circuit is a minimally dependent set, and therefore a circuit cannot contain another circuit. 2. Let X, Y ∈ C(M) where X 6= Y , and e ∈ X ∩ Y . From 1, it follows that X \ Y is non-empty; let f ∈ X \ Y . Assume on the contrary that (X ∪ Y ) − e i...

For any field F, the set of all functions f : V (G) → F whose sum on each maximal independent set is constant forms a vector space over F. In this paper, we show that the dimension can vary depending on the characteristic of the field. We also investigate the dimensions of these vector spaces and show that while some families, such as chordal graphs, have unbounded dimension, other families, su...