ON THE DIAMETER AND CONNECTIVITY OF BIPARTITE KNESER TYPE-k GRAPHS

Let n∈ Z⁺, $n>1$, and k be an integer, 1≤ k ≤ n-1. The graph HT(n,k) is defined as a graph with vertex set V, as all non-empty subsets of Sₙ=\1,2,3,…,n\. It is a bipartite graph with partition (V₁,V₂), in which V₁ contains the k-element subsets of Sₙ and V₂ contains i-element subsets of Sₙ, for 1≤ i ≤ n, \ i≠ k. An edge exists between a vertex U∈ V₁ and a vertex W ∈ V₂ if either U⊂ W or W ⊂ U. In this paper, we established formulae for the number of edges, vertex connectivity, edge connectivity and degree polynomial. Then, we analysed the clique number and diameter of HT(n,k). We also verified that the graphs HT(n,k) and HT(n,n-k) are isomorphic. Degree-based topological indices such as the general Randic connectivity index, the first general Zagreb index and the general sum connectivity index are also computed. Also, we proved that the line graph of HT(n,k) is not a bipartite graph.