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 independence number α ( G ) $\alpha (G)$ and the dissociation diss $\text{diss}(G)$ of a graph $G$ are largest orders induced subgraphs maximum degree at most 0 1, respectively. We consider possible improvements obvious inequality 2 ≥ $2\alpha (G)\ge \text{diss}(G)$ . For connected cubic graphs distinct from K 4 ${K}_{4}$ , we show 5 3 $5\alpha 3\,\text{diss}(G)$ describe rich interesting s...

The analogue of Hadwiger’s conjecture for the immersion order states that every graph G contains Kχ(G) as an immersion. If true, this would imply with n vertices and independence number α K⌈nα⌉ best currently known bound is due to Gauthier, Le Wollan, who recently proved a clique on ⌈χ(G)−43.54⌉ vertices. Their result implies n-vertex ⌈n3.54α−1.13⌉ We improve all α≥3, by showing α≥3 ⌊n2.25α−f(α...

Let G be a finite and simple graph with vertex set V (G), and let f : V (G) → {−1, 1} be a two-valued function. If ∑ x∈N [v] f(x) ≤ 1 for each v ∈ V (G), where N [v] is the closed neighborhood of v, then f is a signed 2independence function onG. The weight of a signed 2-independence function f is w(f) = ∑ v∈V (G) f(v). The maximum of weights w(f), taken over all signed 2-independence functions ...

A 2-factor-plus-triangles graph is the union of two 2-regular graphs G1 and G2 with the same vertices, such that G2 consists of disjoint triangles. Let G be the family of such graphs. These include the famous “cycle-plus-triangles” graphs shown to be 3-choosable by Fleischner and Stiebitz. The independence ratio of a graph in G may be less than 1/3; but achieving the minimum value 1/4 requires ...

In 2014, some scholars showed that every 2-connected claw-free graph G with independence number α(G)≤3 is Hamiltonian one exception of family graphs. If a nontrivial path contains only internal vertices degree two and end not two, then we call it branch. A set S branches called branch cut if delete all edges leading to more components than G. We use bond denote minimal cut. branch-bond has an o...

Let G be a graph with no three independent vertices. How many edges of G can be packed with edge-disjoint copies of Kk? More specifically, let fk(n,m) be the largest integer t such that for any graph with n vertices, m edges, and independence number 2, at least t edges can be packed with edge-disjoint copies ofKk. Turán’s Theorem together with Wilson’s Theorem assert that fk(n,m) = (1−o(1)) 2 4...