Ebrahimi Atani, S., Dolati Pishhesari, S., Khoramdel, M., Sedghi, M. (2018). Total graph of a $0$-distributive lattice. Categories and General Algebraic Structures with Applications, 9(1), 15-27.

Shahabaddin Ebrahimi Atani; Saboura Dolati Pishhesari; Mehdi Khoramdel; Maryam Sedghi. "Total graph of a $0$-distributive lattice". Categories and General Algebraic Structures with Applications, 9, 1, 2018, 15-27.

Ebrahimi Atani, S., Dolati Pishhesari, S., Khoramdel, M., Sedghi, M. (2018). 'Total graph of a $0$-distributive lattice', Categories and General Algebraic Structures with Applications, 9(1), pp. 15-27.

Ebrahimi Atani, S., Dolati Pishhesari, S., Khoramdel, M., Sedghi, M. Total graph of a $0$-distributive lattice. Categories and General Algebraic Structures with Applications, 2018; 9(1): 15-27.

^{}Faculty of Mathematical Sciences, University of Guilan, Rasht, Iran

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.

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