A prime labeling of a simple undirected graph G is to assign unique integer labels from the set {1,2,...,|V(G)|} to each vertex such that any two adjacent vertices in the graph have labels that are relatively prime. studying prime labeling in graphs can help us understand the structure and properties of graphs and prime labeling has potential applications in cryptography and network security. In this paper we investigate when some graphs that are constructed from wheels are prime graphs.

In the context of a simple undirected graph G G , a k k -prime labeling refers to assigning distinct integers from the set { k , k + 1 , & mldr; , divided by V ( G ) divided by + k - 1 } \left\{k,k+1,\ldots ,| V\left(G)| +k-1\right\} to its vertices, such that adjacent vertices in G G are labeled with numbers that are relatively prime to each other. If G G has a k k -prime labeling, we say that G G is a k k -prime graph (k-PG). In this article, we characterize when a graph up to order 6 is a k-PG and characterize when a graph of order 7 is a k-PG whenever k k and k + 1 k+1 are not divisible by 5. Also, we find a lower bound for the independence number of a k-PG. Finally, we study when a cycle is a k-PG.

A prime labeling of a graph G is a map from the vertex set of G, V(G), to the set {1, 2, ..., |V(G)|} such that any two adjacent vertices in the graph G have labels that are relatively prime. In this paper, we discuss when the disjoint union of some graphs is a prime graph.

Let \(X\) be bipartite mixed graph and for a unit complex number \(\alpha\), \(H_\alpha\) be its \(\alpha\)-hermitian adjacency matrix. If \(X\) has a unique perfect matching, then \(H_\alpha\) has a hermitian inverse \(H_\alpha^{-1}\). In this paper we give a full description of the entries of \(H_\alpha^{-1}\) in terms of the paths between the vertices. Furthermore, for \(\alpha\) equals the primitive third root of unity \(\gamma\) and for a unicyclic bipartite graph \(X\) with unique perfect matching, we characterize when \(H_\gamma^{-1}\) is \(\pm 1\) diagonally similar to \(\gamma\)-hermitian adjacency matrix of a mixed graph. Through our work, we have provided a new construction for the \(\pm 1\) diagonal matrix.

A graph G is called semi square stable if alpha(G2) = i(G) where alpha(G2) is the independence number of G2 and i(G) is the independent dominating number of G. A subset S of the vertex set of a graph G is an efficient dominating set if S is an independent set and every vertex of G is either in S or adjacent to exactly one vertex of S. In this paper, we show that every square stable graph has an efficient dominating set and if a graph has an efficient dominating set, then it is semi square stable. We characterize when the join and the corona product of two disjoint graphs are semi square sable graphs and when they have efficient dominating sets.

For a fixed finite group G, the power graph of G was defined to be the simple graph Γ(G) whose vertex set V(Γ(G))=G, and edge set E(Γ(G))={xy: either x=yn or y=xn for some integer n}. In this paper the extreme vertices of the power graph of abelian groups, dihedral groups and dicyclic groups have been characterized.

Let R be a finite commutative ring with nonzero unity and let Z(R) be the zero divisors of R. The total graph of R is the graph whose vertices are the elements of R and two distinct vertices x; y epsilon R are adjacent if x broken vertical bar y epsilon Z(R). The total graph of a ring R is denoted by tau(R). The independence number of the graph tau(R) was found in [11]. In this paper, we again find the independence number of tau(R) but in a different way. Also, we find the independent dominating number of tau(R). Finally, we examine when the graph tau(R) is well-covered.

A graph G is called a well covered graph if every maximal independent set in G is maximum, and co-well covered graph if its complement is a well covered graph. We study some properties of a co-well covered graph and we characterize when the join, the corona product, and cartesian product are co-well covered graphs. Also we characterize when powers of trees and cycles are co-well covered graphs. The line graph of a graph which is co-well covered is also studied.

In this paper it is determined when the line graphs and the middle graphs of some classes of graphs are divisor graphs. Complete characterizations for cycles, trees, complete graphs and complete multipartite graphs whose line graphs (middle graphs) are divisor graphs are obtained. It is also shown that the line graphs and the middle graphs of the cycle permutation graphs are never divisor graphs.

In 1978, Robert Kibler at the National Security Agency in Fort Meade, Maryland published a description of all noncyclic difference sets with $k < 20$. Kibler's decision to stop his extensive computer search for difference sets at block size 19 was motivated partly by the difficult barrier at $k=20$, the difference sets with parameters $(96,20,4)$. In this paper, we announce the completion of the search for all $(96,20,4)$ difference sets, relying on the computer software GAP and the work of numerous authors over the last few decades. The difference sets and the symmetric designs they create are summarized and links are provided to webpages which explicitly list the difference sets. In addition, we use these $(96,20,4)$ difference sets to construct all $(96, 20, 4, 4)$ and $(96, 19, 2, 4)$ partial difference sets and briefly look at the corresponding strongly regular graphs.

The commuting graph of a ring R, denoted by Γ(R), is a graph whose vertices are all non-central elements of R and two distinct vertices x and y are adjacent if and only if xy = yx. Let Zn[i] be the commutative ring of Gaussian integers modulo n, and let Zn[α] be the ring of dual numbers. In this paper we investigate diameters of the commuting graphs of Zn[i] and the commuting graph of Zn[α]. We show that Γ(M(m1 ⊕ m2,Zn[i])) and Γ(M(m1 ⊕ m2,Zn[α])) are connected and diam(Γ(M(m1 ⊕m2, Zn[i]))) = diam(Γ(M(m1 ⊕m2,Zn[α]))) = 3.

In this paper, we find an upper bound of the metric dimension of power of paths and complement of paths. Also, we determine the metric dimension for P 2 n , P 3 n , P 4 n where Pn is a path of length n. Finally, we investigate the metric dimension of certain permutation of paths of odd order. AMS subject classification:

There are 267 nonisomorphic groups of order 64. It was known that 259 of these groups admit (64, 28,12) difference sets. In [4], the author found all (64,28,12) difference sets in 111 groups. In this paper we find all (64,28,12) difference sets in all the remaining groups of order 64 that admit (64, 28,12) difference sets. Also, we find all nonisomorphic symmetric (64, 28, 12) designs that rise from these difference sets. We use these (64,28,12) difference sets to construct all (64,27,10,12) and (64,28,12,12) partial difference sets. Finally, we look at the corresponding strongly regular graphs with parameters (64,27,10,12) and (64,28,12,12).

In this paper, we show that Q(n)(k) is a divisor graph, for n = 2, 3. For n >= 4, we show that Q(n)(k) is a divisor graph iff k >= n - 1. For folded-hypercube, we get FQ(n) is a divisor graph when n is odd. But, if n >= 4 is even integer, then FQ(n) is not a divisor graph. For n >= 5, we show that (FQ(n))(k) is not a divisor graph, where 2 <= k <= [n/2] - 1.

In this paper, we prove that for any tree T, T-2 is a divisor graph if and only if T is a caterpillar and the diameter of T is less than six. For any caterpillar T and a positive integer k >= 1 with diam(T) < 2k, we show that T-k is a divisor graph. Moreover, for a caterpillar T and k >= 3 with diam(T) = 2k or diam(T) = 2k + 1, we show that T-k is a divisor graph if and only if the centers of T have degree two.

The zero-divisor graph of a commutative ring with unity (say R) is a graph whose vertices are the nonzero zero-divisors of this ring, where two distinct vertices are adjacent when their product is zero. This graph is denoted by Gamma(R). In this paper, we study the structure of the zero-divisor graph Gamma(Z(pn) (x)) where p is an odd prime number, Z(p)n is the set of integers modulo p(n), and Z(pn) (x) = {a+bx : a,b is an element of Z(pn) and x(2) = 0}. We find the Independence number of Gamma(Z(pn) (x)).

Let R be a commutative finite principal ideal ring with unity, and let G(R) be the simple graph consisting of nontrivial proper ideals of R as vertices such that two vertices I and J are adjacent if they have nonzero intersection. In this paper we continue the work done by Abu Osba. We calculate the radius, eccentricity, domination number, independence number, geodetic number, and the hull number for this graph. We also determine when G(R) is chordal. Finally, we study some properties of the complement graph of G(R).

Let Pn+1 be the path of order n + 1 on the vertices v(0), v(1), ... , v(n) and P-n+1(k) is the kth power of Pn+1. In this paper, we find the geodetic, hull, and Steiner numbers of P-n+1(k).

There are 267 nonisomorphic groups of order 64. It was known that 259 of these groups admit (64, 28, 12) difference sets and the other eight groups do not admit (64, 28, 12) difference sets. Despite of this result, no research investigates the problem of finding all (64, 28, 12) difference sets in a certain group of order 64. In this paper, we find all (64, 28, 12) difference sets in 111 groups of order 64. 106 of these groups are nonabelian. The other five are Z(16) x Z(4), Z(16) x Z(2)(2), Z(8) x Z(8), Z(8) x Z(4) x Z(2), and Z(8) x Z(2)(3). In these 111 groups we get 74922 nonequivalent (64, 28, 12) difference sets. These difference sets provide at least 105 nonisomorphic symmetric (64, 28, 12) designs. Most of our work have been done by programs using the software GAP.