Papers by Sivagnanam Mutharasu
Journal | MESA, 2020
Let $D = (V, A)$ be a directed graph with $p$ vertices and $q$ arcs. For an integer $k \geq 1$ an... more Let $D = (V, A)$ be a directed graph with $p$ vertices and $q$ arcs. For an integer $k \geq 1$ and for $v \in V(D)$, let $w_k(v) = \sum\limits_{e \in E_k(v)}f(e)$, where $E_k (v)$ is the set of all in-neighborhood arcs which are at distance at most $k$ from $v$. The digraph $D$ is said to be \textit{$E_k$-regular} with regularity $r$ if and only if $\left| E_k(e) \right| = r$ for some integer $r \geq 1$ and for all $e \in A(D)$, where $E_k(e)=E_k(u,v)=\{w \in V(D) : 1 \leq d(w,v) \leq k \}$. An $E_k$-super vertex in-magic total labeling ($E_k$-SVIMTL) is a bijection $f: V(D) \cup A(D) \rightarrow \{ 1,2 , \ldots, p + q \}$ with the property that $f(A(D)) = \{ 1,2,\ldots, q \}$ and for each $v \in V(D)$, $f(v) + w_k(v) = M$ for some positive integer $M$. A digraph that admits an $E_k$-SVIMTL is called an $E_k$-super vertex in-magic total ($E_k$-SVIMT). This paper contains several properties of $E_k$-SVIMTL in digraphs. We obtain a necessary and sufficient condition for the existence ...
Journal of Physics: Conference Series, 2021
Let D be a digraph of order p and size q. For an integer k ⩾ 1 and ν ∈ V(D), let ∑e∈Ek(v)f(e) whe... more Let D be a digraph of order p and size q. For an integer k ⩾ 1 and ν ∈ V(D), let ∑e∈Ek(v)f(e) where Ek (v) is the set of all in-arcs which are at distance at most k from ν. A V k -super vertex in-magic labeling (Vk -SVIML) is an one-to-one onto function f : V(D) ∪ A(D) → {1,2…, P + q} such that f(V(D)) = {1,2…, p} and for every ν ∈ V(D), f(ν) + ω k(ν) = M for some positive integer M. A digraph that admits a V k -SVIML is called V k -super vertex in-magic (V k -SVIM). In this paper, we study some properties of V k -SVIML in digraphs. We characterized the digraphs which are V k -SVIM. Also, we find the magic constant for Ek -regular digraphs. Farther, we characterized the unidirectional cycles and union of unidirectional cycles which are V 2-SMM.
Let D(V, A) be a digraph of order p and sizeq. For an integer k ≥ 1 and forv ∈ V (D), let wk(v) =... more Let D(V, A) be a digraph of order p and sizeq. For an integer k ≥ 1 and forv ∈ V (D), let wk(v) = ∑ e∈Ek(v) f(e), whereEk(v) is the set containing all arcs which are at distance at most k from v. The digraphD is said to beEk-regular with regularityr if and only if |Ek(e)| = r for some integer ≥ 1 and for alle ∈ A(D). A Vk-super vertex out-magic labeling ( Vk-SVOML) is an one-to-one onto function f : V (D)∪A(D) → {1, 2, . . . , p+q} such thatf(V (D)) = {1, 2, . . . , p} and there exists a positive integer M such thatf(v) + wk(v) = M , ∀ v ∈ V (D). A digraph that admits aVk-SVOML is calledVk-super vertex out-magic ( Vk-SVOM). This paper contains several properties of Vk-SVOML in digraphs. We characterized the digraphs which are VkSVOM. Also, the magic constant for Ek-regular graphs has been obtained. Further, we characterized the unidirectional cycles and union of unidirectional cycles which areV2-SVOM. AMS (MOS) Subject Classification Codes: 05C78
In text editors like Notepad and vi editor, we can find a string and replace it with another one.... more In text editors like Notepad and vi editor, we can find a string and replace it with another one. It is not possible to find multiple strings and replace with a single string. This paper reveals other 2 approaches: Finding multiple strings and replacing it with a single string, finding single string and replacing it with multiple strings. Java based string manipulations have been carried out on the contents of the file to implement our functionalities. Thus, our tool provides excellent portability option.
Let G be a graph with p vertices and q edges. An Ek-super vertex magic labeling (Ek-SVML) is a bi... more Let G be a graph with p vertices and q edges. An Ek-super vertex magic labeling (Ek-SVML) is a bijection f : V (G) ∪ E(G) → {1, 2, . . . , p + q} with the property that f(E(G)) = {1, 2, . . . , q} and for each v ∈ V (G), f(v) + wk(v) = M for some positive integer M . For an integer k ≥ 1 and for v ∈ V (G), let wk(v) = ∑ e∈Ek(v) f(e), where Ek(v) is the set of all edges which are at distance at most k from v. The graph G is said to be Ek-regular with regularity r if and only if |Ek(e)| = r for some integer r ≥ 1 and for all e ∈ E(G). A graph that admits an Ek-SVML is called Ek-super vertex magic (Ek-SVM). This paper contain several properties of Ek-SVML in graphs. A necessary and sufficient condition for the existence of Ek-SVML in graphs has been obtained. Also, the magic constant for Ek-regular graphs has been obtained. Further, we establish E2-SVML of some classes of graphs such as cycles, complement of cycles, prism graphs and a family of circulant graphs. MSC: 05C78.
Malaya Journal of Matematik
Let G be a finite and simple (p, q) graph. An one-one onto function f : V (G) ∪ E(G) → {1, 2, 3,.... more Let G be a finite and simple (p, q) graph. An one-one onto function f : V (G) ∪ E(G) → {1, 2, 3,. .. , p + q} is called Vsuper vertex magic graceful labeling if f (V (G)) = {1, 2, 3,. .. , p} and for any vertex v ∈ V (G), ∑ u∈N(v) f (uv)− f (v) = M, where M is a whole number. For an integer k ≥ 1, let E k (v) = {e ∈ E(G) : the distance between e from v is less than or equal to k}. For v ∈ V (G), we define w k (v) = ∑ e∈E k (v) f (e). A V k-super vertex magic graceful labeling (V k-SVMGL) is a one-one function f from V (G) ∪ E(G) onto the set {1, 2, 3,. .. , p + q} such that f (V (G)) = {1, 2, 3,. .. , p} and for any element v ∈ V (G), we have w k (v) − f (v) = M, where M is a whole number. In this paper, we study several properties of V k-SVMGL and we identify an equivalent condition for the E k-regular graphs which admits V k-SVMGL. At last we identify some families of graphs which admit V 2-SVMGL.
Malaya Journal of Matematik
Let G be a simple graph with p vertices and q edges. A V-super vertex magic labeling is a bijecti... more Let G be a simple graph with p vertices and q edges. A V-super vertex magic labeling is a bijection f : V (G) ∪ E(G) → {1, 2,. .. , p + q} such that f (V (G)) = {1, 2,. .. , p} and for each vertex v ∈ V (G), f (v) + ∑ u∈N(v) f (uv) = M for some positive integer M. A V k-super vertex magic labeling (V k-SVML) is a bijection f : V (G) ∪ E(G) → {1, 2,. .. , p + q} with the property that f (V (G)) = {1, 2,. .. , p} and for each v ∈ V (G), f (v) + w k (v) = M for some positive integer M. A graph that admits a V k-SVML is called V k-super vertex magic. This paper contains several properties of V k-SVML in graphs. A necessary and sufficient condition for the existence of V k-SVML in graphs has been obtained. Also, the magic constant for E k-regular graphs has been obtained. Further, we study some classes of graphs such as cycles, complement of cycles, prism graphs and a family of circulant graphs which admit V 2-SVML.
Advanced Studies in Contemporary Mathematics, 2012
A circulant graph is a Cayley graph constructed out of a finite cyclic group Γ and a generating s... more A circulant graph is a Cayley graph constructed out of a finite cyclic group Γ and a generating set A of Γ. Sharp upper bounds are obtained for the domination number, the total domination number and the connected domination number of circulant graphs and exact values for some of the parameters are determined in certain cases. Further, efficient dominating sets of some classes of circulant graphs are also obtained.
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.
A Cayley graph is a graph constructed out of a group and its generating set. In this paper, we de... more A Cayley graph is a graph constructed out of a group and its generating set. In this paper, we define generalized circulant graphs and attempt to find dominating sets and independent dominating sets for the same. Actually we find the values of the domination number, independent domination number and total domination number for generalized circulant graphs. Also it is proved that certain generalized circulant graphs are excellent and in some other cases there are two excellent 2-excellent.
Mathematical Combinatorics (International Book Series), Vol. 4, 2009
Abstract: A Cayley graph is constructed out of a group Γ and its generating set X and it is denot... more Abstract: A Cayley graph is constructed out of a group Γ and its generating set X and it is denoted by C (Γ, X). A Smarandachely n-Cayley graph is defined to be G= ZC (Γ, X), where V (G)= Γ× Zn and E (G)={((x, 0),(y, 1)) a,((x, 1),(y, 2)) a,···,((x, n− 2),(y, n− 1)) a: x, y∈ Γ, a∈ X such that y= x∗ a}. Particularly, a Smarandachely 2-Cayley graph is called as a Bi-Cayley graph, denoted by BC (Γ, X). Necessary and sufficient conditions for the existence of an efficient dominating set and an efficient open dominating set in Bi-Cayley graphs are ...
Discrete Applied Mathematics, Oct 1, 2012
Cayley graphs are graphs constructed out of a finite group Γ and its generating set A. Circulant ... more Cayley graphs are graphs constructed out of a finite group Γ and its generating set A. Circulant graphs are Cayley graphs corresponding to finite cyclic groups. In this paper, the value of the domination number for some circulant graphs is obtained and a corresponding dominating set is also determined. At last a necessary and sufficient condition for a subgroup to be an efficient dominating set in circulant graphs is also obtained.
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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} $.
Mathematical Combinatorics, 2010
Abstract: A Cayley graph is constructed out of a group Γ and its generating set X and it is denot... more Abstract: A Cayley graph is constructed out of a group Γ and its generating set X and it is denoted by С (Γ, X). A Smarandachely n-Cayley graph is defined to be G= ZC (Γ, X), where V (G)= Γ× Zn and E (G)={((x, 0),(y, 1)) a,((x, 1),(y, 2)) a,···,((x, n− 2),(y, n− 1)) a: x, y∈ Γ, a∈ X such that y= x∗ a}. Particularly, a Smarandachely 2-Cayley graph is called as a Bi-Cayley graph, denoted by BС (Γ, X).
Int. J. Open Problems Compt. Math, Jun 1, 2011
Cayley graph is a graph constructed out of a group Γ and a generating set A⊆ Γ. When Γ= Zn, the c... more Cayley graph is a graph constructed out of a group Γ and a generating set A⊆ Γ. When Γ= Zn, the corresponding Cayley graph is called as a circulant graph and denoted by Cir (n, A). In this paper, we attempt to find the total domination number of Cir (n, A) for a particular generating set A of Zn and a minimum total dominating set of Cir (n, A). Further, it is proved that Cir (n, A) is 2-total excellent if and only if n= t| A|+ 1 for some integer t> 0. Keywords: circulant graph, k-total excellent graph, total domination.
Malaya Journal of Matematik
A dominating set S of a graph G is said to be an isolate dominating set of G if the induced subgr... more A dominating set S of a graph G is said to be an isolate dominating set of G if the induced subgraph < S > has at least one isolated vertex [6]. A dominating set S of a graph G is said to be an unique isolate dominating set(UIDS) of G if < S > has exactly one isolated vertex. An UIDS S is said to be minimal if no proper subset of S is an UIDS. The minimum cardinality of a minimal UIDS of G is called the UID number, denoted by γ U 0 (G). This paper includes some properties of UIDS and gives the UID number of paths, complete k-partite graphs and disconnected graphs. Finally, the role played by UIDS in the domination chain has been discussed in detail.
Applied Mathematics Letters, Jan 28, 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.
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Papers by Sivagnanam Mutharasu