A GENERALIZATION OF TOTAL GRAPHS OF MODULES

Let R be a commutative ring, and let M 6= 0 be an R-module with a non-zero proper submodule N , where N? = N − {0}. Let ΓN? (M) denote the (undirected) simple graph with vertices {x ∈M −N |x + x′ ∈ N? for some x 6= x′ ∈M −N}, where distinct vertices x and y are adjacent if and only if x+ y ∈ N?. We determine some graph theoretic properties of ΓN? (M) and investigate the independence number and chromatic number. Mathematics Subject Classification (2010): 13C99, 05C69


Introduction
Throughout, all rings are commutative with non-zero identity and all modules are unitary. Let R be a ring, M = 0 an R-module, and N a non-zero proper submodule of M . The total graph of a commutative ring R, denoted by T (Γ(R)), was introduced by Anderson and Badawi in [3], as the graph with all elements of R as vertices, and two distinct vertices x, y ∈ R are adjacent if and only if x + y ∈ Z(R), where Z(R) denotes the set of zero-divisors of R. The concept of total graphs is a great concept that is usually used in commutative algebra to obtain many interesting graphs in this field. In [1] and [2], A. Abbasi and S. Habibi, gave a generalization of the total graph. They studied in [2]  Let G be a simple graph. If there is a path from any vertex to any other vertex of graph G, then G is said to be connected, and G is said to be totally disconnected This work has been supported by the University of Guilan.

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if there is no path connecting any pair of vertices. For vertices x 1 and x 2 of G, we define d(x 1 , x 2 ) to be the length of a shortest path between x 1 and x 2 (d(x, x) = 0 and d(x 1 , x 2 ) = ∞ if there is no such path). The diameter of G is diam(G) = sup{d(x 1 , x 2 )| x 1 and x 2 are vertices of G}. The girth of G, denoted by gr(G), is the length of its shortest cycle; gr(G) = ∞ if G contains no cycles, in this case, G is called an acyclic graph. A complete graph is one which every two vertices are adjacent. A complete graph with n vertices is denoted by K n . A bipartite graph G is a graph whose vertex set V (G) can be partitioned into disjoint subsets U 1 and U 2 in such a way that each edge of G has one end vertex in U 1 and the other in U 2 . In particular, if E consists of all possible such edges, then G is called a complete bipartite graph and is denoted by K m,n when |U 1 | = m and |U 2 | = n.
For a vertex v of G, deg(v) denotes the degree of v and we set δ(G) := min{deg(v): v is a vertex of G}. A graph G is called k-regular if every vertex has degree k. A subgraph of G is the graph formed by a subset of the vertices and edges of G. Two subgraphs G 1 and G 2 of G are said to be disjoint if G 1 and G 2 have no common vertices and no vertex of G 1 (resp., G 2 ) is adjacent (in G) to any vertex not in G 1 (resp., G 2 ). The union of two graphs G 1 = (V 1 , E 1 ) and G 2 = (V 2 , E 2 ) is the graph The clique number, ω(G), is the greatest integer n ≥ 1 such that K n is a subgraph of G, and ω(G) = ∞ if K n ⊆ G for all n ≥ 1. A matching in a graph G is a set of edges such that no two have a vertex in common.
A spanning matching of a graph is said to be a perfect matching. A star graph S k is the complete bipartite graph K 1,k . A Hamiltonian cycle is a cycle that visits each vertex exactly once. A graph that contains a Hamiltonian cycle is called a Hamiltonian graph. A walk is an alternating sequence of vertices and edges which are incident, that begins and ends with a vertex. A tour is a closed walk that traverses each edge at least once. An Eulerian tour in an undirected graph is a tour that traverses each edge exactly once. If such a tour exists, the graph is called Eulerian. A connected component (or just component) of an undirected graph is a maximal connected induced subgraph. An independent set is a set of vertices in a graph, no two of which are adjacent. That is, it is a set S of vertices such that for every two vertices in S, there is no edge connecting the two. The vertex independence number of G, often called the independence number, is the cardinality of a largest independent vertex set, i.e., the maximum size among all independent vertex sets of G. The independence number is denoted by α(G).
A vertex cover of G is a set of vertices such that each edge of G is incident to at least one vertex of the set. The vertex cover number is the minimum size among all vertex covers in the graph, denoted by β(G). A coloring of a graph is a proper (vertex) coloring with colors such that no two vertices sharing the same edge have the same color. A coloring using k colors is called a (proper) kcoloring. The smallest number of colors needed to color the vertices of G is called its chromatic number and is denoted by χ(G).
The main objective of this paper is to study some properties of the graph Γ N (M ). We also investigate the independence number and chromatic number of the graph Γ N (M ).  ( is not totally disconnected. (2) Let x, y ∈ V (Γ N (M )) be adjacent with x − y ∈ N , then x + y − x + y ∈ N and x + y + x − y ∈ N ; so 2x, 2y ∈ N .  Proof. Let x, m 1 , m 2 , y be distinct vertices of Γ N (M ) with a path x m 1 m 2 y.
Since x+m 1 , m 1 +m 2 and m 2 +y ∈ N , we have x+y = (x+m 1 )+(y +m 2 )−(m 1 + m 2 ) ∈ N . This yields x + y ∈ N * , since x + y = 0 ; so x and y are adjacent. Proof. Without loss of generality, we may assume that x and y are adjacent; so there is a path x x y y in Γ N (M ). Therefore, x and y are adjacent, by Remark 2.6. In Corollary 2.5, the condition "does not equal zero" is necessary.
For instance, in Example 2.3, set x = 5, y = 9, x = 3, y = 11. Then x and y are adjacent, but x and y are not.
) are distinct vertices connected by a path of length 4, then there exists a path of length 2 between them. In particular, there is x m 2 , and we are done.
Proof. By Theorem 2.7, we can reduce every path of length greater than 3 to a path of length at most 3. (a) Let |N (x) ∩ N (y)| = 1. If 2t ∈ N , then x + t, y + t ∈ N ; so x + y + 2t ∈ N .
This yields x + y ∈ N and so x + y = 0 (since x and y are not adjacent). Therefore,  (2) It is clear, since there is a triangular cycle. By Theorem 2.9, the shortest path from u to w is of length 2 or 3.  Proof. Let x ∈ V (Γ N (M )). If x is adjacent to y, then x + y = a ∈ N and hence, y = a − x for some a ∈ N . There are two cases:  (1) If n is an odd integer, then |N | is even. Let 2x ∈ N for some x ∈ V (Γ N (M )). Then 2x = td for some t ∈ Z. Hence x ∈ N is not a vertex. So (2) Assume that n and k are even integers; then d is an even integer. By Theorem 2.19, the degree of every vertex x is |N | or |N | − 1.
(i) Let n = 2 l for some l ∈ N. If d > 2, then there exists at least one vertex x such that 2x / ∈ N . So, the degree of the vertex x is |N |. Note that |N | is an  (3) Let n be an even integer and k be an odd integer. Since |N | is an odd integer and by part (1), 2x / ∈ N for every x ∈ V (Γ N (M )), Γ N (M ) is not Eulerian.
3. Independence number and chromatic number of Γ N (M ) One of the interesting computing problems in graph theory is determining the independence number of a graph. Here we obtain the independence number of Γ N (M ) with some special conditions. It is well-known that α(K n ) = 1. (1) A set is independent if and only if its complement is a vertex cover.
(2) The sum of the size of the largest independent set α(G) and the size of a minimum vertex cover β(G) is equal to the number of vertices in the graph.  For every n ∈ V (Γ N (M )) − {x, −x}, n ∈ P x or −n ∈ P x .
Claim: P x is an independent set in Γ N (M ).
By way of contradiction, suppose there exist n 1 , n 2 ∈ P x that they are adjacent.
Since n 1 , n 2 ∈ P x , so n 1 , n 2 are not adjacent to x. We claim that for every vertex t other than x and −x, t is adjacent to either −x or x (but not to both of them, otherwise, x t (−x), this yields d(x, −x) = 2). Hence, n 1 , n 2 are not adjacent to x, which implies that n 1 , n 2 are adjacent to −x.    Proof.
(1) Let x ∈ V (Γ N (M )). Considering our hypothesis and by the proof of part 1 of Theorem 3.2, for every vertex t other than x and −x, t is adjacent to either −x or x (but not to both of them, otherwise, x t (−x), this implies that d(x, −x) = 2). If t is adjacent to x, then t is not adjacent to −x; so t ∈ P −x , otherwise, t ∈ P x . Hence P x ∪ P −x = V (Γ N (M )). Now we assign color a to elements of P x and color b to elements of P −x . Therefore, χ(Γ N (M )) = 2.