Total graph of a $0$-distributive lattice

Shahabaddin Ebrahimi Atani,Saboura Dolati Pishhesari,Maryam Sedghi,Mehdi Khoramdel

doi:10.29252/cgasa.9.1.15

Shahabaddin Ebrahimi Atani, Saboura Dolati Pishhesari + Show 2 more

Open Access

https://doi.org/10.29252/cgasa.9.1.15

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Abstract

Let £ be a $0$-distributive lattice with the least element $0$, the greatest element $1$, and ${rm Z}(£)$ its set of zero-divisors. In this paper, we introduce the total graph of £, denoted by ${rm T}(G (£))$. It is the graph with all elements of £ as vertices, and for distinct $x, y in £$, the vertices $x$ and $y$ are adjacent if and only if $x vee y in {rm Z}(£)$. The basic properties of the graph ${rm T}(G (£))$ and its subgraphs are studied. We investigate the properties of the total graph of $0$-distributive lattices as diameter, girth, clique number, radius, and the independence number.

Highlights

There has been a lot of activity over the past several years in associating a graph to an algebraic system such as a ring or semiring [1, 3, 5, 8, 9, 11, 14]
We introduce the total graph of a lattice £ with respect to zero-divisor elements of £, denoted by T(G(£))
It is the graph with all elements of £ as vertices, and for distinct x, y ∈ £, the vertices x and y are adjacent if and only if x ∨ y ∈ Z(£)

Summary

Introduction

There has been a lot of activity over the past several years in associating a graph to an algebraic system such as a ring or semiring [1, 3, 5, 8, 9, 11, 14]. * Corresponding author Keywords: Lattice, minimal prime ideal, zero-divisor graph, total graph. We introduce the total graph of a lattice £ with respect to zero-divisor elements of £, denoted by T(G(£)).

Results

Conclusion