Invariants of Bipartite Kneser B type-\MakeLowercase{k} graphs

Abstract

Let $\mathscr{B}_n = \{ \pm x_1, \pm x_2, \pm x_3, \cdots, \pm x_{n-1}, x_n \}$ where $n>1$ is fixed, $x_i \in \mathbb{R}^+$, $i = 1, 2, 3, \cdots, n$ and $x_1 < x_2 < x_3 < \cdots < x_n$. Let $\phi(\mathscr{B}_n)$ be the set of all non-empty subsets $S = \{u_1, u_2,\cdots, u_t\}$ of $\mathscr{B}_n$ such that $|u_1|<|u_2|<\cdots <|u_{t-1}|<u_t $ where $u_t\in \mathbb{R}^+$. Let $\mathscr{B}_n^+ = \{ x_1, x_2, x_3, \cdots, x_{n-1}, x_n \}$. For a fixed $k$, let $V_1$ be the set of $k$-element subsets of $\mathscr{B}_n^+$, $1 \leq k <n$. $V_2= \phi(\mathscr{B}_n)-V_1$. For any $A \in V_2$, let $A^\dagger = \{\lvert x \rvert: x \in A\}$. Define a bipartite graph with parts $V_1$ and $V_2$ and having adjacency as $X \in V_1$ is adjacent to $Y\in V_2$ if and only if $X \subset Y^\dagger$ or $Y^\dagger \subset X$. A graph of this type is called a bipartite Kneser B type-$k$ graph and denoted by $H_B(n,k)$. In this paper, we calculated various graph invariants of $H_B(n,k)$.