%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