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Let L be a lattice, 𝒥(L) be the set of ideals of L and S be a subset of 𝒥 (L). In this paper, we introduce an undirected Cayley graph of L, denoted by ΓL,S with elements of 𝒥 (L) as the vertex set and, for two distinct vertices I and J, I is adjacent to J if and only if there is an element K of S such that I ∨ K = J or J ∨ K = I. We study some basic properties of the graph ΓL,S such as connectivity, girth and clique number. Moreover, we investigate the planarity, outerplanarity and ring graph of ΓL,S.

Given a graph H, a path of length at least two is called an H-path if meets H exactly in its ends. A graph G is a ring graph if each block of G which is not a bridge or a vertex can be constructed inductively by starting from a single cycle and then in each step adding an H-path that meets graph H in the previous step in two adjacent vertices. In this article, we classify all graphs whose line graphs and total graphs are ring graphs.

For a commutative ring R with unity, the associate ring graph, denoted by AG(R), is a simple graph with vertices as nonzero elements of R and two distinct vertices are adjacent if they are associates. The graph AG(R) contains components equal in number to the number of distinct orbits, except for the orbit of an element 0. Moreover, each component is a complete graph. An important finding is that this is a class of strongly perfect graphs. In this article we describe the structure of the associate ring graph of the ring of integers modulo n, denoted by AG(Zn). We carried out computer experiments and provide a program for the same. We further characterize cases in which AG(Zn), its complement AG(Zn)¯, and their line graphs are planar, ring graphs, and outerplanar. We also discuss the properties of the associate ring graph of a commutative ring R with unity.