The Eigensharp Property for Unit Graphs Associated with Some Finite Rings

: Let R be a commutative ring with unity. The unit graph G ( R ) is deﬁned such that the vertex set of G ( R ) is the set of all elements of R , and two distinct vertices are adjacent if their sum is a unit in R . In this paper, we show that for each prime, p , G ( Z p ) and G ( Z 2 p ) are eigensharp graphs. Likewise, we show that the unit graph associated with the ring Z p [ x ] (cid:30) (cid:104) x 2 (cid:105) is an eigensharp graph.


Introduction
Studying rings by associating various graphs with the ring via its algebraic structure has attracted the attention many researchers. Beck [1] introduced the zero-divisor graph; Anderson and Badawi [2] introduced the total graph. Grimaldi [3] defined the unit graph G(Z n ) associated with the finite ring Z n , where the author studied some properties of a graph, such as the Hamilton cycles, covering number, independence number, and chromatic polynomial. The units of a ring play a crucial role in determining the structure of the ring, and many features of a ring can be known from these units. So, it is natural to make a connection between a ring with a graph whose edges have a strong relationship with the units of the ring. The unit graph of a ring is one of such graphs.
In 2010, Ashrafi et al. [4] generalized the unit graph G(Z n ) to G(R) for an arbitrary (commutative) ring R, and considered standard concepts of graph theory such as connectedness, chromatic index, diameter, girth, and planarity of G(R). Akbari et al. [5] studied the unit graph of a noncommutative ring. Maimani et al. [6] showed that the unit graphs is Hamiltonian if and only if the ring R is generated by its units. Heydari and Nikmehr [7] investigated the case when the ring R is a left Artinian ring. Afkhami and Khosh-Ahang [8] studied the unit graphs of rings of polynomials and power series.
A biclique is a complete bipartite subgraph of G. The complete bipartite graphs K 1,n are called stars, denoted by S n . A collection H G = {B 1 , B 2 , . . . , B k } of subgraphs of G is called a biclique partition covering of a graph G if B i is a biclique subgraph for all i = 1, 2, . . . , k, and for every edge e ∈ E(G), there exists exactly one B i ∈ H G , such that e ∈ E(B i ). The biclique partition number of a graph G, denoted by bp(G), is given by bp(G) = min {|H G | : H G and is a biclique partition covering of G}.
One motivation for studying this parameter is to minimize storage space; listing the subgraphs in a minimum complete bipartite decomposition of G never takes more space than the adjacency list representation. Moreover, the biclique partition number has applications in diverse fields of applied science, such as computational complexity, automata and language theories, partial orders, artificial intelligence, and geometry (see, for example, [9][10][11][12][13]). When Graham and Pollak [14] first studied this parameter for the complete graph, they were motivated by a network addressing problem. For more details about graph addressing, please see [15]. The adjacency matrix of G, denoted by A(G), is a square matrix of order |V(G)|, with the ijth entry equaling 1 if v i v j is an edge of G and 0 otherwise. Witsenhausen (see, for example, [14]) showed that for a graph G where a + (G) and a − (G) are the number of positive and negative eigenvalues of the adjacency matrix A(G), respectively. We repeatedly use this fact below. We say that G is an eigensharp graph if bp(G) = max{a + (G), a − (G)}, and it is almost eigensharp if bp(G) = max{a + (G), a − (G)} + 1. Certain families of graphs, including complete graphs K n , complete bipartite graphs K n,m , trees, cycles C n with n = 4 or n = 4k, and various graph products, are eigensharp (see, for example, [16][17][18][19]).
The unit graph G(R) is defined such that the vertex set of G(R) is the set of all elements of the ring R, and two distinct vertices are adjacent if their sum is a unit in R. In this paper, we show that for each prime p, G(Z p ), G(Z 2p ) and G( Z p [x] x 2 ) are eigensharp graphs.

Preliminaries
In this paper, R is assumed to be a commutative ring with unity. An element a is said to be a unit in R if a has a multiplicative inverse. The set U(R) is defined to be the set of all units in R. Moreover, the polynomial ring over Z n is denoted by Z n [x]. In particular, a is a unit in Z n if the greatest common divisor between n and a is equal to 1. For example, Several properties of the unit graph are provided in [4], from which we cite the following Theorem: Theorem 1. [4] Let R be a finite ring. If 2 ∈ U(R), then for every x ∈ U(R), degree (x) = |U(R)| − 1 and for every x ∈ R − U(R), degree (x) = |U(R)|.
All graphs in this paper are finite undirected simple graphs. For a graph G = (V(G), E(G)), the set V(G) denotes the vertex set of G, and E(G) denotes the edge set of G. The degree of a vertex in G is defined as the number of edges emanating from the vertex. A graph G is said to be (n, m)-semiregular if each vertex in G has a degree n or m.
For a simple graph G, the adjacency matrix A(G) is a symmetric matrix with real eigenvalues such that the algebraic multiplicity is equal to geometric multiplicity for each eigenvalue. We refer to it as multiplicity. It can be proved that a + (G) > 0 and a − (G) > 0 for any non-null graph G.
The multiplicity of an eigenvalue λ i is the number of linearly independent eigenvectors associated with it. If λ i , 1 ≤ i ≤ j are the distinct eigenvalues of the adjacency matrix A(G) For example, The join of two graphs G and H, denoted by G ∨ H, is the graph with vertex set G ∨ H is a complete bipartite graph if both G and H are independent vertices. The following Theorem was proved in [20]. Theorem 2. [20] Suppose that G and H are two regular graphs. Then, a − (G ∨ H) = a − (G) + a − (H) + 1 and a + (G ∨ H) = a + (G) + a + (H) − 1. Consequently, if each G and H are eigensharp graphs with bp(G) = a − (G) and bp(H) = a − (H), then G ∨ H is an eigensharp graph.

Unit Graph Associated with Rings Z p and Z 2p
In this section, we obtain the biclique partition number of G(Z p ), and we prove that G(Z p ) is an eigensharp graph.
Theorem 3. For each prime p, the graph G(Z p ) is eigensharp.
Proof. If p = 2 and 3, then G(Z p ) is isomorphic to P 2 and P 3 , respectively. Hence, Hence, for p = 2 or 3, G(Z p ) is an eigensharp graph. Now, for p ≥ 5, let V = {0, 1, . . . , p − 1} and E = {e r,s : r + s ∈ U(Z p )} be the vertex set and the edge set of where 0 is adjacent to each nonzero element in Z p , and H is a (p − 3)-regular graph. It has been found and from several computations for different p's that A(H) is a (p − 1) × (p − 1) matrix that has the form The enteries of A(H) are all 1, except 0 on the main and secondary diagonals. Notably, the first p−1 2 columns are linearly independent. The Hence, B i is isomorphic to K 2,(p−1)−2i . Note that no pair of edges of H belongs to a common B i (X i , Y i ), and See [21], Theorem 3.

Theorem 4. The graph G Z 2p is eigensharp.
Proof. Note that the graph G Z 2p is a graph with 2p vertices. Suppose that the vertex set is V G Z 2p = {0, 1, 2, . . . , 2p − 1}. Then, the two distance vertices in G Z 2p are adjacent if their sum is an odd number less than 2p and not equal to p.

Now, the adjacency matrix of
is the adjacency matrix of the complete graph K p . Using Remark 1, we claim that the spectrum To prove this claim, we notice that On the other hand, let H G(Z 2p ) = {S 2k : 0 ≤ k ≤ p − 1} be the set of p disjoint stars in G Z 2p generated by the vertices 2k, 0 ≤ k ≤ p − 1. Then, H G(Z 2p ) is a biclique partition of cardinality p. Hence, the graph G Z 2p is eigensharp.

Unit Graph Associated with the Ring Z p [x]
x 2 In this section, we consider the ring Z p [x] x 2 ) is eigensharp. We denote the graph G( Z p [x] x 2 ) by G p (x 2 ). Let s = p 2 − p and J p be a p × p matrix, where all entries are ones; let 1 p be a p × 1 matrix, where all entries are ones, N p be the zero matrix of size p × p, and 0 p be the zero matrix of size p × 1. For m = 1, 2, . . . , p−1 2 define the partition matrix F (m) as the s × 1 matrix such that all the submatrices entries are 0 p , except for the mth row, which is the submatrix 1 p , and the (p − m) row is the submatrix −1 p . Furthermore, for r = 2, 3, . . . , p−1 2 defines the partition matrix H (r) as the s × 1, where all the submatrices are 0 p , except the first and last rows are the submatrix 1 p , and the rth and (p − r) rows are the submatrix −1 p . For example, if p = 11, then, x 2 . Then, a + bX is a unit if and only if a is a unit in Z p . Thus, ). Thus, v is adjacent with each vertex in G p (x 2 ), except {a + bX, (p − a), (p − a) + X, . . . , (p − a) + (p − 1)X}, i.e., v has a degree p 2 − (p + 1) = p(p − 1) − 1. Now, we consider the subgraph W of G p (x 2 ) induced by V(W) = V(G p (x 2 ))\T. Let m = (p(p − 1) − p − 1). Then, W is an m-regular graph with It is clear that G p (x 2 ) is isomorphic to T ∨ W. Mainly, we show that W is an eigensharp graph with bp(W) = a − (W) and, by Theorem 2, G p (x 2 ) is an eigensharp.
Therefore, the characteristic polynomial of A(W) is E(F i ) = {e r,s : r ∈ ℘, s ∈ }.
Then, F i is isomorphic to K 2p,p(p−2i−1) with no pair of edges of E(W), which belongs to a common F i and Now, consider the disjoint stars S j+tx in B j generated by the vertices Then, is a biclique partition of W with cardinality p 2 + (p − 1) 2 − 1. Therefore, W is an eigensharp graph with bp(W) = p 2 + (p − 1) 2 − 1, which implies that G p (x 2 ) is an eigensharp graph.

Conclusions
In this study, for each prime p; we proved that the graphs G(Z p ), G(Z 2p ) and x 2 are eigensharps. We showed that G(Z p ) is isomorphic to a graph K 1 ∨ H, where H is a certain subgraph of G(Z p ) and G Z p [x] x 2 is isomorphic to T ∨ W, where T is a certain independent set of G Z p [x] x 2 and W is a certain subgraph of G Z p [x] x 2 . Then, the adjacency matrices for H and W were studied to show that a − (H) = bp(H) and a − (W) = bp(W), which yields, by Theorem 2, that both graphs G(Z p ) and G Z p [x] x 2 are eigensharps. The spectrum of the graph A(G(Z 2p )) was found to demonstrate that bp(G(Z 2p )) ≥ p. We also described a biclique partition for G(Z 2p ) with cardinality p; we hence concluded that G(Z 2p ) is eigensharp.
Finally, we raise the following question: Does the eigensharp property hold for Z p n , Z pq and Z p [x] x n ? We have attempted several examples to answer this question, but our research is still ongoing.

Conflicts of Interest:
The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.