Dr. Kumar Abhishek currently serves as Assistant Professor (SG) in the Department of Mathematics, School of Engineering, Coimbatore Campus. His areas of research include Graph Theory.

Publications

Publication Type: Journal Article

Year of Publication

Publication Type

Title

2016

Journal Article

K. Abhishek, “ARC dimension of a digraph”, Proceedings of the Jangjeon Mathematical Society, vol. 19, pp. 107-114, 2016.[Abstract]

The results of Harary, Norman, and Cartwright on point-bases in finite digraphs to point- and arc-bases in infinite digraphs was extended by Acharya et.al, [1] by introducing the notion of arc bases of digraphs as follows: in a digraph D = (X,U), not necessarily finite, an arc (x,y) εU is reachable from a vertex u if there exists a directed walk W that originates from u and contains (x,y). A subset S ⊆ X is an arc-reaching set of D if for every arc (x,y) there exists a diwalk W originating at a vertex u ε S and containing (x,y) and an arc-basis as a minimal arc-reaching set. One of the main results reported in [1] is that all the arc bases of any finite digraph D have the same cardinality which led to the introduction of the notion of arc-dimension of D, denoted σ(D), as the cardinality of an arc basis of D. In this article we establish the upper and lower bounds on σ(D)+σ(D→), σ(D)+σ(Dc) and establish some related results.

K. Abhishek, “Some New Classes of Harmonious Graphs”, Proceedings of the Jangjeon Mathematical Society, vol. 19, pp. 293–299, 2016.[Abstract]

For a finite graph G of order p and size q, let V(G) and E(G) denote its vertex and edge set respectively. A harmonious labeling of a connected graph G is an injective function λ: V(G) → Z_{q} such that the induced edge function λ∗: V(G) → Z_{q} defined as λ∗(xy) = λ (x) + λ(y)]modq for each edge xy ϵ E(G) is a bijection whenever G is not a tree. If G is a tree, exactly one label may be used on two vertices. In this note we report some new classes of harmonious graphs.
More »»

2015

Journal Article

K. Abhishek, “Set-Valued Graphs: A Survey”, Journal of Discrete Mathematical Sciences and Cryptography, vol. 18, pp. 55-80, 2015.[Abstract]

The problem of set valuation of a graph requires both the vertices and edges of an undirected simple graph G to be labeled with subsets of a nonempty set. The label of an edge uv of G is obtained as the symmetric difference of the subsets assigned to the vertices u and v of G. A graph G is said to be set-valued if there exists an assignment of subsets of a nonempty set on the vertices of G such that the following two conditions holds:(i) all the subsets on the vertices are distinct and, (ii) all the subsets on the edges are distinct. The objective of this article is to organize and summarize much of the work done on set-valued graphs since its inception in 1983. Many open problems and conjectures are included. We explore new directions with regards to the enumeration of set-valued graphs.

K. Abhishek, “An Application of Arc-Reaching Sets in Social-Network Analysis”, Journal of Discrete Mathematical Sciences and Cryptography, vol. 18, pp. 409-415, 2015.[Abstract]

K. Abhishek and Ganesan, A., “Detour distance pattern of a graph”, International Journal of Pure and Applied Mathematics, vol. 87, pp. 719-728, 2013.[Abstract]

K. Abhishek and Germina, K. A., “Out Set-Magic Digraphs”, Journal of Discrete Mathematical Sciences and Cryptography, vol. 7, no. 1, pp. 4173 – 4184, 2013.[Abstract]

Abstract Motivated by the papers of Peay [1], Acharya [2], and SedlaÄek [3], we introduce out set-magic indexer of digraphs: Let X be a nonempty set, 2 X denote the power-set of X. As in [2], given a digraph D with p vertices and q arcs, no-self loops, and parallel arcs, is labeled by assigning to each vertex an element from the set 2 X . An arc (x, y) from a vertex x to y is labeled with f âŠ•(x, y) = f (x) âŠ• f (y), where f (x) and f (y) are the values assigned to x and y, and â€œ âŠ• â€ is the symmetric difference of the sets. Such an assignment is called a set-indexer if f and f âŠ• are injective. A set-indexer f of a digraph D is called an out set-magic indexer if, U xâˆ¼e f âŠ•(e) = X, for all xâˆˆ V(D). A digraph admitting an out set-magic indexer is called an out set-magic digraph. In this paper we give some necessary conditions for a digraph to admit an out set-magic indexer, provide the sharp bounds on the size of a digraph admitting an out set-magic indexer. We prove there exist no digraph admitting a unique out set-magic indexer. Also, we give a construction of an out set-magic indexed digraph from a directed path, directed cycle, tournament, inspoken wheel, and directed wind mill, etc., thereby showing their embeddings.

K. Abhishek, “Set-Valued Graphs II”, Journal of Fuzzy Set Valued Analysis, vol. 2013 , p. 16, 2013.[Abstract]

A set-indexer of a graph G is an assignment of distinct subsets of a finite set of n elements to the vertices of the graph, where the edge values are obtained as the symmetric differences of the set assigned to their end vertices which are also distinct. A set-indexer is called set-sequential if sets on the vertices and edges are distinct and together form the set of all nonempty subsets of A set-indexer called set-graceful if all the nonempty subsets of are obtained on the edges. A graph is called set-sequential (set-graceful) if it admits a set-sequential (set-graceful) set-indexer. In the recent literature the notion of set-indexer has appeared as set-coloring. While obtaining in general a `good' characterization of a set-sequential (set-graceful) graphs remains a formidable open problem ever since the notion was introduced by Acharya in 1983, it becomes imperative to recognize graphs which are set-sequential (set-graceful). In particular, the problem of characterizing set-sequential trees was raised raised by Acharya in 2010. In this article we completely characterize the set-sequential caterpillars of diameter five.

B. D. Acharya, Germina, K. A., Abhishek, K., and Slater, P. J., “Some New Results on Set-Graceful and Set-Sequential Graphs”, Journal of Combinatorics & System Sciences, vol. 37, no. 2-4, p. 229, 2012.[Abstract]

A set-indexer of a given graph G = (V, E) is an assignment f of distinct nonempty subsets of a finite nonempty 'ground set' X = {x1, x2,...,xn} of car dinality n, where 2X denotes the power set of X, to the vertices of G so that the values f âŠ•(e), e = uv âˆˆ E; obtained as the symmetric differences f(u) âŠ• f(v) of the subsets f(u) and f(v) of X, are all distinct. It is well known that every graph admits a set-indexer. A function f : Vâˆª E â†’ Y = 2X â€“ {âˆ…} is called a set-sequential labeling of G = (V, E) if it is a bijection and for all uv âˆˆE, f(u) âŠ• f(v) = f(uv): A graph is called set-sequential if it admits a set-sequential labeling. A set-indexer f of a graph G = (V, E) is called a set-graceful labeling of G if there exists a nonempty ground set X such that fâŠ•(E) = 2X âˆ’ {âˆ…} and G is setgraceful if it admits a set-graceful labeling. In this article we provide characterization of m copies of K2 , mK2 , that are set-sequential and the friendship graphs C3m, consisting of m triangles attached at one common vertex that are set-graceful. It is also established that for every set X of odd cardinality there is a set-sequential tree of diameter four.

B. D. Acharya, Abhishek, K., and Germina, K. A., “Hypergraphs of Minimal Arc Bases in A Digraph”, Journal of Combinatorics & System Sciences, vol. 37, no. 2-4, pp. 307-319, 2012.[Abstract]

In a digraph D = (X, U), not necessarily finite, an arc (x, y) âˆˆ U is reachable from a vertex u if there exists a directed walk W that originates from u and contains (x, y). A subset S âŠ† X is an arc-reaching set of D if for every arc (x, y) there exists a diwalk W originating at a vertex u âˆˆ S and containing (x, y). A minimal arc-reaching set is an arc-basis. S is a point-reaching set if for every vertex v there exists a diwalk W to v originating at a vertex u âˆˆ S. A minimal point-reaching set is a point-basis. A study of hypergraphs formed by minimal arc bases of a digraph is the main objective of this paper.

K. A. Germina and Abhishek, K., “Set-Valued Graphs”, Journal of Fuzzy Set Valued Analysis, p. 17, 2012.

2010

Journal Article

K. A. Germina and Abhishek, K., “Kernels and Grundy functions of hypergraphs.”, South East Asian Journal of Mathematics and Mathematical Sciences, vol. 9, no. 1, pp. 55–63, 2010.

2009

Journal Article

B. D. Acharya, Germina, K. A., Abhishek, K., Rao, S. B., and Zaslavsky, T., “Point-and arc-reaching sets of vertices in a digraph”, Indian Journal of Mathematics, vol. 51, no. 3, pp. 597–609, 2009.[Abstract]

Abstract: In a digraph $ D=(X,\ mathcal {U}) $, not necessarily finite, an arc $(x, y)\ in\ mathcal {U} $ is reachable from a vertex $ u $ if there exists a directed walk $ W $ that originates from $ u $ and contains $(x, y) $. A subset $ S\ subseteq X $ is an arc-reaching set of $ D $ if for every arc $(x, y) $ there exists a diwalk $ W $ originating at a vertex $ u\ in S $ and containing $(x, y) $. A minimal arc-reaching set is an arc-basis. $ S $ is a point-reaching set if for every vertex $ v $ there exists a diwalk $ W $ to $ v $ riginating at a vertex $ u\ in S $
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2009

Journal Article

B. K. Jose, Germina, K. A., and Abhishek, K., “On some open problems of stable sets and domination in hypergraphs”, Un-published, 2009.

2008

Journal Article

K. A. Germina, Abhishek, K., and Princy, K. L., “Further results on set-valued graphs”, Journal of Discrete Mathematical Sciences and Cryptography, vol. 11, no. 5, pp. 559–566, 2008.[Abstract]

Acharya [2] introduced the notion of set-valuation as: G=(V, E) be a (p, q) graph, X be a nonempty set of cardinality n and 2 X denote the set of all subsets of X. A set-indexer of G is an injective set-valued function f : V(G) → 2 X such that the function f ⊕: E(G) → 2 X −{∅} defined by f ⊕(uv)=f (v) ⊕ f (v) for every uv∈E(G) is also injective, where ⊕ is the symmetric difference of sets. A(p,q)-graph G=(V,E) is set-graceful if it admits a set-graceful labeling which is a set-valued injection f: V→2 X such that the function f ⊕ : E → 2 X defined by f ⊕ (uv)=f (u) ⊕ f (u) for all uv∈E is such that f ⊕ (E) ≔ {f ⊕(uv) : uv∈E}=2 X −{∅} and set-sequential if it admits a set-valued injection f : V∪E → 2 X , called a set-sequential labeling of G, such that f (V∪E) ≔ {f (x) : x∈V∪E}=2 X −{∅}. In this paper, we contribute two new necessary conditions for a graph to be set-sequential. In addition, we characterize stars that are set-sequential and also establish that certain specifically structured trees such as binary trees and bistars are not set-sequential. Also we prove that wheel is not set-sequential. Acharya in a personal communication to the first author in April 2007 during her visit to ISI, Delhi conjectured that “Tree TP obtained by pegging the whole of a binary tree T to a pivot P through an extra edge, called the hanger, joining P to the ‘top’ vertex of T (which is its centre as such), the new tree TP is set-sequential!”. We prove the conjecture for uniform binary trees.
More »»

Dr. Kumar Abhishek currently serves as Assistant Professor (SG) in the Department of Mathematics, School of Engineering, Coimbatore Campus. His areas of research include Graph Theory.

## Publications

The results of Harary, Norman, and Cartwright on point-bases in finite digraphs to point- and arc-bases in infinite digraphs was extended by Acharya et.al, [1] by introducing the notion of arc bases of digraphs as follows: in a digraph D = (X,U), not necessarily finite, an arc (x,y) εU is reachable from a vertex u if there exists a directed walk W that originates from u and contains (x,y). A subset S ⊆ X is an arc-reaching set of D if for every arc (x,y) there exists a diwalk W originating at a vertex u ε S and containing (x,y) and an arc-basis as a minimal arc-reaching set. One of the main results reported in [1] is that all the arc bases of any finite digraph D have the same cardinality which led to the introduction of the notion of arc-dimension of D, denoted σ(D), as the cardinality of an arc basis of D. In this article we establish the upper and lower bounds on σ(D)+σ(D→), σ(D)+σ(Dc) and establish some related results.

More »»For a finite graph G of order p and size q, let V(G) and E(G) denote its vertex and edge set respectively. A harmonious labeling of a connected graph G is an injective function λ: V(G) → Z

_{q}such that the induced edge function λ∗: V(G) → Z_{q}defined as λ∗(xy) = λ (x) + λ(y)]modq for each edge xy ϵ E(G) is a bijection whenever G is not a tree. If G is a tree, exactly one label may be used on two vertices. In this note we report some new classes of harmonious graphs. More »»The problem of set valuation of a graph requires both the vertices and edges of an undirected simple graph G to be labeled with subsets of a nonempty set. The label of an edge uv of G is obtained as the symmetric difference of the subsets assigned to the vertices u and v of G. A graph G is said to be set-valued if there exists an assignment of subsets of a nonempty set on the vertices of G such that the following two conditions holds:(i) all the subsets on the vertices are distinct and, (ii) all the subsets on the edges are distinct. The objective of this article is to organize and summarize much of the work done on set-valued graphs since its inception in 1983. Many open problems and conjectures are included. We explore new directions with regards to the enumeration of set-valued graphs.

More »»A set-indexer [1] of the graph G is an assignment of distinct subsets of a finite set X to the vertices of the graph, where the sets on the edges are obtained as the symmetric differences of the sets assigned to their end vertices which are also distinct. A set-indexer is called a set-sequential if sets on the vertices and edges are distinct and together form the set of all nonempty subsets of X. A set-indexer is called a set-graceful if all the nonempty subsets of X are obtained on the edges. A graph is called set-sequential (set-graceful) if it admits a set-sequential (set-graceful) labeling. The objective of this note is to report some new results, open problems and conjectures in relation to set-indexed graphs. © 2015, © Taru Publications.

More »»URL: http://www.tandfonline.com/doi/abs/10.1080/09720529.2013.867637#abstract

<p>In this article we present an application of the notion of arc-bases, introduced by Acharya et.al, in 2009, to the social network analysis. © 2015, © Taru Publications.</p>

More »»For a simple connected graph G = (V,E), let M ⊇ V and u ∈ V. The M-detour distance pattern of G is the set fM(u) = {D(u, v) : v ∈ M}. If fM is injective function, then the set M is a detour distance pattern distinguishing set (or, ddpd-set in short) of G. A graph G is defined as detour distance pattern distinguishing (or, ddpd-) graph if it admits a ddpd-set. The objective of this article is to initiate the study of graphs that admit marker set M for which fM is injective. This article establishes some general results on ddpd-graphs. © 2013 Academic Publications, Ltd.

More »»Abstract Motivated by the papers of Peay [1], Acharya [2], and SedlaÄek [3], we introduce out set-magic indexer of digraphs: Let X be a nonempty set, 2 X denote the power-set of X. As in [2], given a digraph D with p vertices and q arcs, no-self loops, and parallel arcs, is labeled by assigning to each vertex an element from the set 2 X . An arc (x, y) from a vertex x to y is labeled with f âŠ•(x, y) = f (x) âŠ• f (y), where f (x) and f (y) are the values assigned to x and y, and â€œ âŠ• â€ is the symmetric difference of the sets. Such an assignment is called a set-indexer if f and f âŠ• are injective. A set-indexer f of a digraph D is called an out set-magic indexer if, U xâˆ¼e f âŠ•(e) = X, for all xâˆˆ V(D). A digraph admitting an out set-magic indexer is called an out set-magic digraph. In this paper we give some necessary conditions for a digraph to admit an out set-magic indexer, provide the sharp bounds on the size of a digraph admitting an out set-magic indexer. We prove there exist no digraph admitting a unique out set-magic indexer. Also, we give a construction of an out set-magic indexed digraph from a directed path, directed cycle, tournament, inspoken wheel, and directed wind mill, etc., thereby showing their embeddings.

More »»A set-indexer of a graph G is an assignment of distinct subsets of a finite set of n elements to the vertices of the graph, where the edge values are obtained as the symmetric differences of the set assigned to their end vertices which are also distinct. A set-indexer is called set-sequential if sets on the vertices and edges are distinct and together form the set of all nonempty subsets of A set-indexer called set-graceful if all the nonempty subsets of are obtained on the edges. A graph is called set-sequential (set-graceful) if it admits a set-sequential (set-graceful) set-indexer. In the recent literature the notion of set-indexer has appeared as set-coloring. While obtaining in general a `good' characterization of a set-sequential (set-graceful) graphs remains a formidable open problem ever since the notion was introduced by Acharya in 1983, it becomes imperative to recognize graphs which are set-sequential (set-graceful). In particular, the problem of characterizing set-sequential trees was raised raised by Acharya in 2010. In this article we completely characterize the set-sequential caterpillars of diameter five.

More »»A set-indexer of a given graph G = (V, E) is an assignment f of distinct nonempty subsets of a finite nonempty 'ground set' X = {x1, x2,...,xn} of car dinality n, where 2X denotes the power set of X, to the vertices of G so that the values f âŠ•(e), e = uv âˆˆ E; obtained as the symmetric differences f(u) âŠ• f(v) of the subsets f(u) and f(v) of X, are all distinct. It is well known that every graph admits a set-indexer. A function f : Vâˆª E â†’ Y = 2X â€“ {âˆ…} is called a set-sequential labeling of G = (V, E) if it is a bijection and for all uv âˆˆE, f(u) âŠ• f(v) = f(uv): A graph is called set-sequential if it admits a set-sequential labeling. A set-indexer f of a graph G = (V, E) is called a set-graceful labeling of G if there exists a nonempty ground set X such that fâŠ•(E) = 2X âˆ’ {âˆ…} and G is setgraceful if it admits a set-graceful labeling. In this article we provide characterization of m copies of K2 , mK2 , that are set-sequential and the friendship graphs C3m, consisting of m triangles attached at one common vertex that are set-graceful. It is also established that for every set X of odd cardinality there is a set-sequential tree of diameter four.

More »»In a digraph D = (X, U), not necessarily finite, an arc (x, y) âˆˆ U is reachable from a vertex u if there exists a directed walk W that originates from u and contains (x, y). A subset S âŠ† X is an arc-reaching set of D if for every arc (x, y) there exists a diwalk W originating at a vertex u âˆˆ S and containing (x, y). A minimal arc-reaching set is an arc-basis. S is a point-reaching set if for every vertex v there exists a diwalk W to v originating at a vertex u âˆˆ S. A minimal point-reaching set is a point-basis. A study of hypergraphs formed by minimal arc bases of a digraph is the main objective of this paper.

More »»Abstract: In a digraph $ D=(X,\ mathcal {U}) $, not necessarily finite, an arc $(x, y)\ in\ mathcal {U} $ is reachable from a vertex $ u $ if there exists a directed walk $ W $ that originates from $ u $ and contains $(x, y) $. A subset $ S\ subseteq X $ is an arc-reaching set of $ D $ if for every arc $(x, y) $ there exists a diwalk $ W $ originating at a vertex $ u\ in S $ and containing $(x, y) $. A minimal arc-reaching set is an arc-basis. $ S $ is a point-reaching set if for every vertex $ v $ there exists a diwalk $ W $ to $ v $ riginating at a vertex $ u\ in S $ More »»

Acharya [2] introduced the notion of set-valuation as: G=(V, E) be a (p, q) graph, X be a nonempty set of cardinality n and 2 X denote the set of all subsets of X. A set-indexer of G is an injective set-valued function f : V(G) → 2 X such that the function f ⊕: E(G) → 2 X −{∅} defined by f ⊕(uv)=f (v) ⊕ f (v) for every uv∈E(G) is also injective, where ⊕ is the symmetric difference of sets. A(p,q)-graph G=(V,E) is set-graceful if it admits a set-graceful labeling which is a set-valued injection f: V→2 X such that the function f ⊕ : E → 2 X defined by f ⊕ (uv)=f (u) ⊕ f (u) for all uv∈E is such that f ⊕ (E) ≔ {f ⊕(uv) : uv∈E}=2 X −{∅} and set-sequential if it admits a set-valued injection f : V∪E → 2 X , called a set-sequential labeling of G, such that f (V∪E) ≔ {f (x) : x∈V∪E}=2 X −{∅}. In this paper, we contribute two new necessary conditions for a graph to be set-sequential. In addition, we characterize stars that are set-sequential and also establish that certain specifically structured trees such as binary trees and bistars are not set-sequential. Also we prove that wheel is not set-sequential. Acharya in a personal communication to the first author in April 2007 during her visit to ISI, Delhi conjectured that “Tree TP obtained by pegging the whole of a binary tree T to a pivot P through an extra edge, called the hanger, joining P to the ‘top’ vertex of T (which is its centre as such), the new tree TP is set-sequential!”. We prove the conjecture for uniform binary trees. More »»

Scopus Author ID:57190224416Researcher ID: C-5099-2013ORCID ID:orcid.org/0000-0001-9322-5214## Related

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