%0 Journal Article
%T Total graph of a $0$-distributive lattice
%J Categories and General Algebraic Structures with Applications
%I Shahid Beheshti University
%Z 2345-5853
%A Ebrahimi Atani, Shahabaddin
%A Dolati Pishhesari, Saboura
%A Khoramdel, Mehdi
%A Sedghi, Maryam
%D 2018
%\ 07/01/2018
%V 9
%N 1
%P 15-27
%! Total graph of a $0$-distributive lattice
%K Lattice
%K minimal prime ideal
%K zero-divisor graph
%K total graph
%R 10.29252/cgasa.9.1.15
%X 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.
%U https://cgasa.sbu.ac.ir/article_50749_c43feee35e55c325b3f13fa98313523d.pdf