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

Abstract

Let ${\mathcal{B}}_n=\{\pm x_1,\pm x_2,\pm x_3,\cdots ,\pm x_{n-1},x_n\}$ 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 $\varphi ({\mathcal{B}}_n)$ be the set of all non-empty subsets $S=\{u_1,u_2,\cdots ,u_t\}$ of ${\mathcal{B}}_n$ such that $|u_1|<|u_2|<\cdots <|u_{t-1}|<u_t$ where $u_t\in {\mathbb{R}}^{+}$. Let ${\mathcal{B}}_n^{+}=\{x_1,x_2,x_3,\cdots ,x_{n-1},x_n\}$. For a fixed $k$, let $V_1$ be the set of $k$-element subsets of ${\mathcal{B}}_n^{+}$, $1\le k<n$. $V_2=\varphi ({\mathcal{B}}_n)-V_1$. For any $A\in V_2$, let $A^{†}=\{|x|:x\in A\}$. 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 $X\subset Y^{†}$ or $Y^{†}\subset X$. 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 $H_B(n,k)$.

15 pages, 2 figures