In this paper, we investigate the existence of nontrivial solutions for the equation γ(G□H) = γ(G... more In this paper, we investigate the existence of nontrivial solutions for the equation γ(G□H) = γ(G) γ(H) fixing one factor. For the complete bipartite graphs K m,n ; we characterize all nontrivial solutions when m = 2, n ≥ 3 and prove the nonexistence of solutions when m, n ≥ 3. In addition, it is proved that the above equation has no nontrivial solution if H is one of the graphs obtained from C n , the cycle of length n, either by adding a vertex and one pendant edge joining this vertex to any v ∈ V(C n), or by adding one chord joining two alternating vertices of C n .
Let R be a commutative ring with zero-divisor set Z(R). The total graph of R, denoted by T (Γ(R))... more Let R be a commutative ring with zero-divisor set Z(R). The total graph of R, denoted by T (Γ(R)), is the simple (undirected) graph with vertex set R where two distinct vertices are adjacent if their sum lies in Z(R). This work considers minimum zero-sum k-flows for T (Γ(R)). Both for |R| even and the case when |R| is odd and Z(G) is an ideal of R it is shown that T (Γ(R)) has a zero-sum 3-flow, but no zero-sum 2-flow. As a step towards resolving the remaining case, the total graph T (Γ(Zn)) for the ring of integers modulo n is considered. Here, minimum zero-sum k-flows are obtained for n = p r q s (where p and q are primes, r and s are positive integers). Minimum zero-sum k-flows as well as minimum constant-sum k-flows in regular graphs are also investigated. This conjecture is known as the zero-sum conjecture (ZSC). In the same paper, the validity of ZSC is verified for bipartite graphs. Moreover, it is shown that if G is an r-regular graph with r even and r ≥ 4, then G admits a zero-sum 3-flow. Further study of zero-sum flows in regular graphs [3] reveals that if G is an r-regular graph, with r ≥ 3, then G admits a zero-sum 7-flow. Those results
Let Γ(R) be the zero divisor graph for a commutative ring with identity. The k-domination number ... more Let Γ(R) be the zero divisor graph for a commutative ring with identity. The k-domination number and the 2-packing number of Γ(R), where R is an Artinian ring, are computed. k-dominating sets and 2-packing sets for the zero divisor graph of the ring of Gaussian integers modulo n, Γ(Zn[i]), are constructed. The center, the median, the core, as well as the automorphism group of Γ(Zn[i]) are determined. Perfect zero divisor graphs Γ(R) are investigated.
LetR be a commutative ring with nonzero unity. Let Z(R) be the set of all zerodivisors ofR. The t... more LetR be a commutative ring with nonzero unity. Let Z(R) be the set of all zerodivisors ofR. The total graph of R, denoted byT (Γ(R)), is the simple graph with vertex set R and two distinct verticesx andy are adjacent if their sumx + y ∈ Z(R). Several authors presented various generalizations for T (Γ(R)). This article surveys research conducted on T (Γ(R)) and its generalizations. A historical review of literature is given. Further p roperties ofT (Γ(R)) are also studied. Many open problems are presented for further rese arch.
Let R be a commutative ring with unity. The total graph of R, T (Γ(R)), is the simple graph with ... more Let R be a commutative ring with unity. The total graph of R, T (Γ(R)), is the simple graph with vertex set R and two distinct vertices are adjacent if their sum is a zero-divisor in R. Let Reg(Γ(R)) and Z(Γ(R)) be the subgraphs of T (Γ(R)) induced by the set of all regular elements and the set of zero-divisors in R, respectively. We determine when each of the graphs T (Γ(R)), Reg(Γ(R)), and Z(Γ(R)) is locally connected, and when it is locally homogeneous. When each of Reg(Γ(R)) and Z(Γ(R)) is regular and when it is Eulerian.
Let R be a commutative ring with zero-divisor set Z(R). The total graph of R, denoted by T((R)),... more Let R be a commutative ring with zero-divisor set Z(R). The total graph of R, denoted by T((R)), is the simple (undirected) graph with vertex set R where two distinct vertices are adjacent if their sum lies in Z(R). This work considers minimum zero-sum k-flows for T((R)). Both for jRj even and the case when jRj is odd and Z(G) is an ideal of R it is shown that T((R)) has a zero-sum 3-flow, but no zero-sum 2-flow. As a step towards resolving the remaining case, the total graph T((Zn)) for the ring of integers modulo n is considered. Here, minimum zero-sum k-flows are obtained for n = prqs (where p and q are primes, r and s are positive integers). Minimum zero-sum k-flows as well as minimum constant-sum k-flows in regular graphs are also investigated.
In this paper, we investigate the existence of nontrivial solutions of the equation γ(G□U)=γ(G)γ(... more In this paper, we investigate the existence of nontrivial solutions of the equation γ(G□U)=γ(G)γ(H) fixing one factor. We characterize all nontrivial solutions when the fixed graph G is the complete bipartite graph K 2,n ,n≥3 and prove the nonexistence of solutions when G=K m,n ,m,n≥3. In addition, it is proved that the above equation has no nontrivial solution if G is one of the graphs obtained from C n , the cycle of length n, either by adding a vertex and one pendant edge joining this vertex to any v∈V(C n ), or by adding one chord joining two alternating vertices of C n .
The line graph for the complement of the zero divisor graph for the ring of Gaussian integers mod... more The line graph for the complement of the zero divisor graph for the ring of Gaussian integers modulo n is studied. The diameter, the radius and degree of each vertex are determined. Complete characterization of Hamiltonian, Eulerian, planer, regular, locally H and locally connected
Discussiones Mathematicae - General Algebra and Applications, 2014
Let Γ(R) be the zero divisor graph for a commutative ring with identity. The k-domination number ... more Let Γ(R) be the zero divisor graph for a commutative ring with identity. The k-domination number and the 2-packing number of Γ(R), where R is an Artinian ring, are computed. k-dominating sets and 2-packing sets for the zero divisor graph of the ring of Gaussian integers modulo n, Γ(Z n [i]), are constructed. The center, the median, the core, as well as the automorphism group of Γ(Z n [i]) are determined. Perfect zero divisor graphs Γ(R) are investigated.
Let Γ(ℤn[i]) be the zero divisor graph for the ring of the Gaussian integers modulo n. Several pr... more Let Γ(ℤn[i]) be the zero divisor graph for the ring of the Gaussian integers modulo n. Several properties of the line graph of Γ(ℤn[i]), L(Γ(ℤn[i])) are studied. It is determined when L(Γ(ℤn[i])) is Eulerian, Hamiltonian, or planer. The girth, the diameter, the radius, and the chromatic and clique numbers of this graph are found. In addition, the domination number of L(Γ(ℤn[i])) is given when n is a power of a prime. On the other hand, several graph invariants for Γ(ℤn[i]) are also determined.
Let γ(G) denote the domination number of a graph G and let Cn□G denote the cartesian product of C... more Let γ(G) denote the domination number of a graph G and let Cn□G denote the cartesian product of Cn, the cycle of length n⩾3, and G. In this paper, we are mainly concerned with the question: which connected nontrivial graphs satisfy γ(Cn□G)=γ(Cn)γ(G)? We prove that this equality can only hold if n≡1 (mod3). In addition, we characterize graphs which satisfy this equality when n=4 and provide infinite classes of graphs for general n≡1 (mod3).
Let G be a Steinhaus graph generated by the the n-long string ej = (0, . . ., 0,1, 0,...0) where ... more Let G be a Steinhaus graph generated by the the n-long string ej = (0, . . ., 0,1, 0,...0) where 1 is placed in the jth position. The degree sequence as well as the automorphism group of G will be identified.
In this paper, we investigate the existence of nontrivial solutions for the equation γ(G□H) = γ(G... more In this paper, we investigate the existence of nontrivial solutions for the equation γ(G□H) = γ(G) γ(H) fixing one factor. For the complete bipartite graphs K m,n ; we characterize all nontrivial solutions when m = 2, n ≥ 3 and prove the nonexistence of solutions when m, n ≥ 3. In addition, it is proved that the above equation has no nontrivial solution if H is one of the graphs obtained from C n , the cycle of length n, either by adding a vertex and one pendant edge joining this vertex to any v ∈ V(C n), or by adding one chord joining two alternating vertices of C n .
Let R be a commutative ring with zero-divisor set Z(R). The total graph of R, denoted by T (Γ(R))... more Let R be a commutative ring with zero-divisor set Z(R). The total graph of R, denoted by T (Γ(R)), is the simple (undirected) graph with vertex set R where two distinct vertices are adjacent if their sum lies in Z(R). This work considers minimum zero-sum k-flows for T (Γ(R)). Both for |R| even and the case when |R| is odd and Z(G) is an ideal of R it is shown that T (Γ(R)) has a zero-sum 3-flow, but no zero-sum 2-flow. As a step towards resolving the remaining case, the total graph T (Γ(Zn)) for the ring of integers modulo n is considered. Here, minimum zero-sum k-flows are obtained for n = p r q s (where p and q are primes, r and s are positive integers). Minimum zero-sum k-flows as well as minimum constant-sum k-flows in regular graphs are also investigated. This conjecture is known as the zero-sum conjecture (ZSC). In the same paper, the validity of ZSC is verified for bipartite graphs. Moreover, it is shown that if G is an r-regular graph with r even and r ≥ 4, then G admits a zero-sum 3-flow. Further study of zero-sum flows in regular graphs [3] reveals that if G is an r-regular graph, with r ≥ 3, then G admits a zero-sum 7-flow. Those results
Let Γ(R) be the zero divisor graph for a commutative ring with identity. The k-domination number ... more Let Γ(R) be the zero divisor graph for a commutative ring with identity. The k-domination number and the 2-packing number of Γ(R), where R is an Artinian ring, are computed. k-dominating sets and 2-packing sets for the zero divisor graph of the ring of Gaussian integers modulo n, Γ(Zn[i]), are constructed. The center, the median, the core, as well as the automorphism group of Γ(Zn[i]) are determined. Perfect zero divisor graphs Γ(R) are investigated.
LetR be a commutative ring with nonzero unity. Let Z(R) be the set of all zerodivisors ofR. The t... more LetR be a commutative ring with nonzero unity. Let Z(R) be the set of all zerodivisors ofR. The total graph of R, denoted byT (Γ(R)), is the simple graph with vertex set R and two distinct verticesx andy are adjacent if their sumx + y ∈ Z(R). Several authors presented various generalizations for T (Γ(R)). This article surveys research conducted on T (Γ(R)) and its generalizations. A historical review of literature is given. Further p roperties ofT (Γ(R)) are also studied. Many open problems are presented for further rese arch.
Let R be a commutative ring with unity. The total graph of R, T (Γ(R)), is the simple graph with ... more Let R be a commutative ring with unity. The total graph of R, T (Γ(R)), is the simple graph with vertex set R and two distinct vertices are adjacent if their sum is a zero-divisor in R. Let Reg(Γ(R)) and Z(Γ(R)) be the subgraphs of T (Γ(R)) induced by the set of all regular elements and the set of zero-divisors in R, respectively. We determine when each of the graphs T (Γ(R)), Reg(Γ(R)), and Z(Γ(R)) is locally connected, and when it is locally homogeneous. When each of Reg(Γ(R)) and Z(Γ(R)) is regular and when it is Eulerian.
Let R be a commutative ring with zero-divisor set Z(R). The total graph of R, denoted by T((R)),... more Let R be a commutative ring with zero-divisor set Z(R). The total graph of R, denoted by T((R)), is the simple (undirected) graph with vertex set R where two distinct vertices are adjacent if their sum lies in Z(R). This work considers minimum zero-sum k-flows for T((R)). Both for jRj even and the case when jRj is odd and Z(G) is an ideal of R it is shown that T((R)) has a zero-sum 3-flow, but no zero-sum 2-flow. As a step towards resolving the remaining case, the total graph T((Zn)) for the ring of integers modulo n is considered. Here, minimum zero-sum k-flows are obtained for n = prqs (where p and q are primes, r and s are positive integers). Minimum zero-sum k-flows as well as minimum constant-sum k-flows in regular graphs are also investigated.
In this paper, we investigate the existence of nontrivial solutions of the equation γ(G□U)=γ(G)γ(... more In this paper, we investigate the existence of nontrivial solutions of the equation γ(G□U)=γ(G)γ(H) fixing one factor. We characterize all nontrivial solutions when the fixed graph G is the complete bipartite graph K 2,n ,n≥3 and prove the nonexistence of solutions when G=K m,n ,m,n≥3. In addition, it is proved that the above equation has no nontrivial solution if G is one of the graphs obtained from C n , the cycle of length n, either by adding a vertex and one pendant edge joining this vertex to any v∈V(C n ), or by adding one chord joining two alternating vertices of C n .
The line graph for the complement of the zero divisor graph for the ring of Gaussian integers mod... more The line graph for the complement of the zero divisor graph for the ring of Gaussian integers modulo n is studied. The diameter, the radius and degree of each vertex are determined. Complete characterization of Hamiltonian, Eulerian, planer, regular, locally H and locally connected
Discussiones Mathematicae - General Algebra and Applications, 2014
Let Γ(R) be the zero divisor graph for a commutative ring with identity. The k-domination number ... more Let Γ(R) be the zero divisor graph for a commutative ring with identity. The k-domination number and the 2-packing number of Γ(R), where R is an Artinian ring, are computed. k-dominating sets and 2-packing sets for the zero divisor graph of the ring of Gaussian integers modulo n, Γ(Z n [i]), are constructed. The center, the median, the core, as well as the automorphism group of Γ(Z n [i]) are determined. Perfect zero divisor graphs Γ(R) are investigated.
Let Γ(ℤn[i]) be the zero divisor graph for the ring of the Gaussian integers modulo n. Several pr... more Let Γ(ℤn[i]) be the zero divisor graph for the ring of the Gaussian integers modulo n. Several properties of the line graph of Γ(ℤn[i]), L(Γ(ℤn[i])) are studied. It is determined when L(Γ(ℤn[i])) is Eulerian, Hamiltonian, or planer. The girth, the diameter, the radius, and the chromatic and clique numbers of this graph are found. In addition, the domination number of L(Γ(ℤn[i])) is given when n is a power of a prime. On the other hand, several graph invariants for Γ(ℤn[i]) are also determined.
Let γ(G) denote the domination number of a graph G and let Cn□G denote the cartesian product of C... more Let γ(G) denote the domination number of a graph G and let Cn□G denote the cartesian product of Cn, the cycle of length n⩾3, and G. In this paper, we are mainly concerned with the question: which connected nontrivial graphs satisfy γ(Cn□G)=γ(Cn)γ(G)? We prove that this equality can only hold if n≡1 (mod3). In addition, we characterize graphs which satisfy this equality when n=4 and provide infinite classes of graphs for general n≡1 (mod3).
Let G be a Steinhaus graph generated by the the n-long string ej = (0, . . ., 0,1, 0,...0) where ... more Let G be a Steinhaus graph generated by the the n-long string ej = (0, . . ., 0,1, 0,...0) where 1 is placed in the jth position. The degree sequence as well as the automorphism group of G will be identified.
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