Jayakumar C, Sreekumar K. G., Manilal K., Ismail Naci Cangul
{"title":"Invariants of Bipartite Kneser B type-\\MakeLowercase{k} graphs","authors":"Jayakumar C, Sreekumar K. G., Manilal K., Ismail Naci Cangul","doi":"arxiv-2409.09317","DOIUrl":null,"url":null,"abstract":"Let $\\mathscr{B}_n = \\{ \\pm x_1, \\pm x_2, \\pm x_3, \\cdots, \\pm x_{n-1}, x_n\n\\}$ where $n>1$ is fixed, $x_i \\in \\mathbb{R}^+$, $i = 1, 2, 3, \\cdots, n$ and\n$x_1 < x_2 < x_3 < \\cdots < x_n$. Let $\\phi(\\mathscr{B}_n)$ be the set of all\nnon-empty subsets $S = \\{u_1, u_2,\\cdots, u_t\\}$ of $\\mathscr{B}_n$ such that\n$|u_1|<|u_2|<\\cdots <|u_{t-1}|<u_t $ where $u_t\\in \\mathbb{R}^+$. Let\n$\\mathscr{B}_n^+ = \\{ x_1, x_2, x_3, \\cdots, x_{n-1}, x_n \\}$. For a fixed $k$,\nlet $V_1$ be the set of $k$-element subsets of $\\mathscr{B}_n^+$, $1 \\leq k\n<n$. $V_2= \\phi(\\mathscr{B}_n)-V_1$. For any $A \\in V_2$, let $A^\\dagger =\n\\{\\lvert x \\rvert: x \\in A\\}$. Define a bipartite graph with parts $V_1$ and\n$V_2$ and having adjacency as $X \\in V_1$ is adjacent to $Y\\in V_2$ if and only\nif $X \\subset Y^\\dagger$ or $Y^\\dagger \\subset X$. A graph of this type is\ncalled a bipartite Kneser B type-$k$ graph and denoted by $H_B(n,k)$. In this\npaper, we calculated various graph invariants of $H_B(n,k)$.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"7 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Combinatorics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.09317","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
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}|