Cayley subspace sum graph of vector spaces

Year 2023, Volume: 33 Issue: 33, 1 - 17, 09.01.2023

G. Kalaımurugan S. Gopınath T. Tamızh Chelvam

https://doi.org/10.24330/ieja.1195466

Abstract

Let $\mathbb{V}$ be a finite dimensional vector space over the field $\mathbb{F}$. Let $S(\mathbb{V})$ be the set of all subspaces of $\mathbb{V}$ and $\mathbb{A}\subseteq S^*(\mathbb{V})=S(\mathbb{V})\backslash\{0\}.$ In this paper, we define the Cayley subspace sum graph of $\mathbb{V},$ denoted by Cay$(S^*(\mathbb{V}),\mathbb{A}), $ as the simple undirected graph with vertex set $S^*(\mathbb{V})$ and two distinct vertices $X$ and $Y$ are adjacent if $X+Z=Y$ or $Y+Z=X$ for some $Z\in \mathbb{A}$. Having defined the Cayley subspace sum graph, we study about the connectedness, diameter and girth of several classes of Cayley subspace sum graphs Cay$(S^*(\mathbb{V}), \mathbb{A})$ for a finite dimensional vector space $\mathbb{V}$ and $\mathbb{A}\subseteq S^*(\mathbb{V})=S(\mathbb{V})\backslash\{0\}.$

Keywords

Cayley sum graph, vector space, subspace, diameter, girth, planar

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