On the diameter and connectivity of bipartite Kneser type-k graphs

Document Type : Original Manuscript


1 Department of Mathematics, University college, University of Kerala, India

2 Department of mathematics, University of Kerala

3 Professor, Department of Mathematics, University College, University of Kerala, Thiruvananthapuram, Kerala



Let $n\in \mathbb{Z}^{+}$, $n>1$, and $k$ be an integer, $1\leq k \leq n-1$. The graph $H_{T}(n,k)$ is defined as a graph with vertex set $V$, as all non-empty subsets of $\mathcal{S}_n=\{1,2,3,\ldots,n\}$. It is a bipartite graph with partition $(V_{1},V_{2})$, in which $V_{1}$ contains the $k$-element subsets of $\mathcal{S}_n$ and $V_{2}$ contains $i$-element subsets of $\mathcal{S}_n$, for $1\le i \le n, \ i\neq k$. An edge exists between a vertex $U\in V_{1}$ and a vertex $W \in V_{2}$ if either $U\subset W$ or $W \subset U$. 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 $H_T(n,k)$. We also verified that the graphs $H_{T}(n,k)$ and $H_{T}(n,n-k)$ 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 $H_T(n,k)$ is not a bipartite graph.