The Topological Connectivity of the Bipartite Graph's Independence Complex

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Let (∁) signify a simplicial complex, the topological connectivity of ∁ is donated by ζ (∁), which equals ∁+2; where ζ is a function from the class of graphs to the set Z^+={0,1,2,…,∞}. Suppose that the function ζ possesses the following characteristics: 1. ζ(K_0 )=0 2. For any graph G(V,E), there exists, an edge {x,y} in G such that ζ(G(V,E)-{x,y})≥ζ(G(V,E)) where, (G(V,E) -{x,y}) means: to remove the edge {x,y} from the graph G(V,E)), and ζ(G(V,E)-Δ({x,y}))≥ζ(G(V,E))-1 where, (G(V,E)-Δ ({x,y}) is equal to remove the vertices x,y and all their neighbors. then μ(Γ(G(V,E))≥ζ(G(V,E)) Let〖 ζ〗_0 denote the maximal function ζ that satisfying the aforementioned constraints. Eli Berger [4] proved that μ(Γ(G(V,E))≥ζ_0 (G(V,E)) for trees and for complements of chordal graphs. Kazuhiro Kawamura [3] proved the same theorem for chordal graphs, and Yousef Methkal Abd Algani [1] proved it for circular-arc graph: Let G(V,E) be a circular-arc graph, if ζ_0 (G(V,E))≤2,then μ(Γ(G(V,E)))≤2. In the current chapter we will prove the following theorem: Let G(V,E) be a bipartite graph, if ζ_0 (G(V,E))≤2,then μ(Γ(G(V,E)))≤2.

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