Papers by Tamizh Chelvam T
Graphs and Combinatorics
Let G be a finite abelian group, written additively, and H a subgroup of G. The subgroup sum grap... more Let G be a finite abelian group, written additively, and H a subgroup of G. The subgroup sum graph Γ G,H is the graph with vertex set G, in which two distinct vertices x and y are joined if x + y ∈ H \ {0}. These graphs form a fairly large class of Cayley sum graphs. Among cases which have been considered previously are the prime sum graphs, in the case where H = pG for some prime number p. In this paper we present their structure and a detailed analysis of their properties. We also consider the simpler graph Γ + G,H , which we refer to as the extended subgroup sum graph, in which x and y are joined if x + y ∈ H: the subgroup sum is obtained by removing from this graph the partial matching of edges having the form {x, −x} when 2x = 0. We study perfectness, clique number and independence number, connectedness, diameter, spectrum, and domination number of these graphs and their complements. We interpret our general results in detail in the prime sum graphs.
Let R be a commutative ring with identity and let Nil(R) be the ideal of all 2 nilpotent elements... more Let R be a commutative ring with identity and let Nil(R) be the ideal of all 2 nilpotent elements of R. Let I(R) = {I : I is a non-trivial ideal of R and there exists a 3 non-trivial ideal J such that IJ ⊆ Nil(R)}. The nil-graph of ideals of R is defined as the 4 simple undirected graph AG N (R) whose vertex set is I(R) and two distinct vertices I and 5 J are adjacent if and only if IJ ⊆ Nil(R). In this paper, we study the planarity and genus of 6 AG N (R). In particular, we have characterized all commutative Artin rings R for which the 7 genus of AG N (R) is either zero or one.
Let R be a commutative ring with identity and let Nil(R) be the ideal of all 2 nilpotent elements... more Let R be a commutative ring with identity and let Nil(R) be the ideal of all 2 nilpotent elements of R. Let I(R) = {I : I is a non-trivial ideal of R and there exists a 3 non-trivial ideal J such that IJ ⊆ Nil(R)}. The nil-graph of ideals of R is defined as the 4 simple undirected graph AG N (R) whose vertex set is I(R) and two distinct vertices I and 5 J are adjacent if and only if IJ ⊆ Nil(R). In this paper, we study the planarity and genus of 6 AG N (R). In particular, we have characterized all commutative Artin rings R for which the 7 genus of AG N (R) is either zero or one.
Journal of Combinatorial Mathematics and Combinatorial Computing
Let R be a noncommutative ring with identity and Z(R) * be the non-zero zero-divisors of R. The d... more Let R be a noncommutative ring with identity and Z(R) * be the non-zero zero-divisors of R. The directed zero-divisor graph Γ(R) of R is a directed graph with vertex set Z(R) * and for distinct vertices x and y of Z(R) * , there is a directed edge from x to y if and only if xy = 0 in R. S.P. Redmond has proved that for a finite commutative ring R, if Γ(R) is not a star graph, then the domination number of the zero-divisor graph Γ(R) equals the number of distinct maximal ideals of R. In this paper, we prove that such a result is true for the noncommutative ring M2(F), where F is a finite field. Using this we obtain a class of graphs for which all six fundamental domination parameters are equal.
Journal of Combinatorial Mathematics and Combinatorial Computing
Let R be a commutative ring with identity and A * (R) be the set of non-zero ideals with non-zero... more Let R be a commutative ring with identity and A * (R) be the set of non-zero ideals with non-zero annihilators. The annihilatingideal graph of R is defined as the graph AG(R) with the vertex set A * (R) and two distinct vertices I1 and I2 are adjacent if and only if I1I2 = (0). In this paper, we study some connections between the graph-theoretic properties of AG(R) and algebraic properties of commutative ring R.
Let R be a commutative ring with identity and A * (R) the set of non-zero ideals with non-zero an... more Let R be a commutative ring with identity and A * (R) the set of non-zero ideals with non-zero annihilators. The annihilating-ideal graph of R is defined as the graph AG(R) with the vertex set A * (R) and two distinct vertices I 1 and I 2 are adjacent if and only if I 1 I 2 = (0). In this paper, we examine the presence of cut vertices and cut sets in the annihilating-ideal graph of a commutative Artinian ring and provide a partial classification of the rings in which they appear. Using this, we obtain the vertex connectivity of some annihilating-ideal graphs.
The directed zero-divisor graph is a graph constructed out of a non-commutative ring R and its no... more The directed zero-divisor graph is a graph constructed out of a non-commutative ring R and its non-zero zero-divisors Z(R) *. The zero-divisor graph Γ(R) of a non-commutative ring R is a directed graph with vertex set Z(R) * , where for distinct vertices x and y of Z(R) * there is a directed edge from x to y if and only if xy = 0 in R. In this paper, we attempt to find the various domination parameters for the directed zero-divisor graph constructed out of M 2 (Z p), where p is a prime number.
In this paper, we study about the intersection graph of gamma sets in the zero-divisor graph of t... more In this paper, we study about the intersection graph of gamma sets in the zero-divisor graph of the commutative ring Z n. The intersection graph of gamma sets is a graph with vertex set as the collection of all gamma sets of the zero-divisor graph of Z n and two distinct vertices A and B are adjacent if and only if A ∩ B ̸ = ∅. This graph is denoted by I Γ (Z n). We investigate the interplay between the graph theoretic properties of I Γ (Z n) and the ring theoretic properties of Z n and we attempt to find the various domination parameters for the intersection graph of gamma sets in the zero-divisor graph of Z n .
Ars Combinatoria Waterloo Then Winnipeg, 2010
A set D of vertices of a graph G is a dominating set if every vertex of V-D is adjacent to at lea... more A set D of vertices of a graph G is a dominating set if every vertex of V-D is adjacent to at least one vertex in D. The domination number ) (G γ is the minimum cardinality of a dominating set of G. A subset of V-D, which is also a dominating set of G is called the inverse dominating set of G with respect to D. The inverse domination number of G, ) (G γ ′ is the minimum cardinality of the inverse dominating sets. In this paper, some graphs with equal domination and inverse domination numbers are identified.
International Journal of Algebra and Statistics, 2014
Iranian Journal of Mathematical Sciences and Informatics, Nov 15, 2014
Paul Erdos defined the concept of coprime graph and studied about cycles in coprime graphs. In th... more Paul Erdos defined the concept of coprime graph and studied about cycles in coprime graphs. In this paper this concept is generalized and a new graph called Generalized coprime graph is introduced. Having observed certain basic properties of the new graph it is proved that the chromatic number and the clique number of some generalized coprime graphs are equal.
Discrete Mathematics, Algorithms and Applications, 2012
Journal of Advanced Research in Pure Mathematics, 2011
Abstract. In this paper, we obtain necessary and sufficient conditions for the existence of an ef... more Abstract. In this paper, we obtain necessary and sufficient conditions for the existence of an efficient dominating set in the Cartesian product of two cycles and in the Cartesian product of three cycles. When $2 n+ 1$ is prime and $ k_i $'s are multiples of $2 n+ 1$, we obtain all efficient dominating sets in the Cartesian product of $ n $ cycles $\ square _ {i= 1}^{n} C_ {k_i} $.
Journal of Advanced Research in Pure Mathematics, 2010
Abstract. We introduce the concept of a morphism between fuzzy sets, which enables us to define t... more Abstract. We introduce the concept of a morphism between fuzzy sets, which enables us to define the category of fuzzy sets. Next, we give elementwise characterization of some special morphisms in this category which will be useful for later studies. We prove that unlike the category of sets, this category is not balanced. Also, we prove that this category has equalizers, coequalizers, intersections, images and weak inverse images.
Resonance, 2001
Addendum to “A result concerning the stability of some difference equations and its applications”... more Addendum to “A result concerning the stability of some difference equations and its applications” by Hark-Mahn Kim, Vol. 112, No. 3, pp. 453–462 ............. ... Generalized (m, n) bi-ideals of a near-ring ................................... ....................................... T Tamizh Chelvam and S Jayalakshmi 479 ... Limits of rank 4 Azumaya algebras and applications to desingularization........ ............... ..........................................TE Venkata Balaji 485 ... A general theorem characterizing some absolute summability methods ......... .............................................................WT Sulaiman 543
International Journal of Combinatorics, 2012
A subset D of the vertex set of a graph G, is a dominating set if every vertex in V − D is adjace... more A subset D of the vertex set of a graph G, is a dominating set if every vertex in V − D is adjacent to at least one vertex in D. The domination number γ G is the minimum cardinality of a dominating set of G. A subset of V − D, which is also a dominating set of G is called an inverse dominating set of G with respect to D. The inverse domination number γ G is the minimum cardinality of the inverse dominating sets. characterized connected graphs G with γ G γ G n, where n is the number of vertices in G. It is the purpose of this paper to give a complete characterization of graphs G with minimum degree at least two and γ G γ G n − 1.
Applied Mathematics Letters, 2012
Efficient open dominating sets in bipartite Cayley graphs are characterized in terms of covering ... more Efficient open dominating sets in bipartite Cayley graphs are characterized in terms of covering projections. Necessary and sufficient conditions for the existence of efficient open dominating sets in certain circulant Harary graphs are given. Chains of efficient dominating sets, and of efficient open dominating sets, in families of circulant graphs are described as an application.
In this paper two coincidence point theorems in complete metric spaces for two pairs of single an... more In this paper two coincidence point theorems in complete metric spaces for two pairs of single and multi-valued mappings have been established.
Abstract. In this paper, we obtain necessary and sufficient conditions for the existence of an ef... more Abstract. In this paper, we obtain necessary and sufficient conditions for the existence of an efficient dominating set in the Cartesian product of two cycles and in the Cartesian product of three cycles. When $2 n+ 1$ is prime and $ k_i $'s are multiples of $2 n+ 1$, we obtain all efficient dominating sets in the Cartesian product of $ n $ cycles $\ square _ {i= 1}^{n} C_ {k_i} $.
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Papers by Tamizh Chelvam T