CONNECTED MAJORITY DOMINATION VERTEX CRITICAL GRAPHS
CONNECTED MAJORITY DOMINATION VERTEX CRITICAL GRAPHS
J. Joseline Manora, T. M. Vairavel
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Abstract
In this article, how the removal of a single vertex from a graph G can change the Connected Majority Domination number is surveyed for any graph G. A graph is Connected Domination Critical if the removal of any vertex decreases or increases its Connected Majority Domination Number. This paper gives examples and properties of CMD vertex critical graphs. There are two types namely CVR and UVR with respect to CMD sets of a graph. Also the vertex classification. V 0 (G), V − (G) and V + are CM CM CM studied, characterisation theorems of these vertex classification are determined.
Keywords
Majority vertex critical, Connected Majority vertex critical, CV RCM , UV RCM .
RESULTS ON THE INVERSE MAJORITY DOMINATION AND MAJORITY INDEPENDENCE NUMBER OF A GRAPH
RESULTS ON THE INVERSE MAJORITY DOMINATION AND MAJORITY INDEPENDENCE NUMBER OF A GRAPH
J. J. Manora, S. Vignesh
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Abstract
Inthisarticle,therelationshipbetweenInverseMajorityDominationnumber γ−1(G) and Majority Independence number βM(G) of a graph G is discussed for some M classes of graphs. In particular, γ−1(G) and βM(G) for cubic and cubic bipartite graph M are studied with examples. Also characterization theorem for this relation and some results are determined.
Keywords
Majority Dominating Set, Inverse Majority Domination Number, Majority Independence Number , Cubic Bipartite Graphs.
TOTAL VERTEX IRREGULARITY STRENGTH OF INTERVAL GRAPHS
TOTAL VERTEX IRREGULARITY STRENGTH OF INTERVAL GRAPHS
A. Rana
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Abstract
A labeling of a graph is a mapping that maps some set of graph elements to a set of numbers (usually positive integers). For a simple graph G = (V,E) with vertex set V and edge set E, a labeling φ : V ∪ E → {1, 2, ..., k} is called total k-labeling. The associated vertex weight of a vertex x ∈ V (G) under a total k-labeling φ is defined as wt(x) = φ(x)+ E φ(xy) where N(x) is the set of neighbors of the vertex x. A total y∈N(x) k-labeling is defined to be a vertex irregular total labeling of a graph G, if wt(x) ̸= wt(y) holds for every two different vertices x and y of G. The minimum k for which a graph G has a vertex irregular total k-labeling is called the total vertex irregularity strength of G, tvs(G). In this paper, total vertex irregularity strength of interval graphs is studied. In particular, an efficient algorithm is designed to compute tvs of proper interval graphs and bounds of tvs are presented for interval graphs.
Keywords
Interval graphs, vertex irregular total labeling, total vertex irregularity strength, design of algorithms.
TRIPLE CONNECTED ETERNAL DOMINATION IN GRAPHS
TRIPLE CONNECTED ETERNAL DOMINATION IN GRAPHS
G. Mahadevan, T. Ponnuchamy, S. Avadayappan
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Abstract
The concept of Triple connected domination number was introduced by G. Mahadevan et. al., in [10]. The concept of eternal domination in graphs was introduced by W. Goddard., et. al., in [3]. The dominating set S0(⊆ V (G)) of the graph G is said to be an eternal dominating set, if for any sequence v1, v2, v3, . . . vk of vertices, there exists a sequence of vertices u1, u2, u3, . . . uk with ui ∈ Si−1 and ui equal to or adjacent to vi, such that each set Si = Si−1 −{ui}∪{vi} is dominating set in G. The minimum cardinal- ity taken over the eternal dominating sets in G is called the eternal domination number of G and it is denoted by γ∞(G). In this paper we introduce another new concept Triple connected eternal domination in graph. The eternal dominating set S0(⊆ V (G)) of the graph G is said to be a triple connected eternal dominating set, if each dominating set Si is triple connected. The minimum cardinality taken over the triple connected eternal dominating sets in G is called the triple connected eternal domination number of G and it is denoted by γtc,∞(G). We investigate this number for some standard graphs and obtain many results with other graph theoretical parameters.
Keywords
Triple connected domination number, Eternal domination in graphs, Triple connected eternal domination number of graphs.
DIAMETRAL PATHS IN EXTENDED TRANSFORMATION GRAPHS
DIAMETRAL PATHS IN EXTENDED TRANSFORMATION GRAPHS
P. Garg, M. Sharma
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Abstract
In a graph, diametral path is shortest path between two vertices which has length equal to diameter of the graph. Number of diametral paths plays an important role in computer science and civil engineering. In this paper, we introduce the concept of extended transformation graphs. There are 64 extended transformation graphs. We obtain number of diametral paths in some of the extended transformation graphs and we also study the semi-complete property of these extended transformation graphs. Further, a program is given for obtaining number of diametral paths.
Keywords
Diameter, diametral path, transformation graph, semi-complete graph.
KEY DISTRIBUTION USING GRAPHS FOR SECRET SHARING
KEY DISTRIBUTION USING GRAPHS FOR SECRET SHARING
R. Malhotra, N. Chandramowliswaran
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Abstract
Key distribution for secret sharing is the core important aspect of any secure cryptosystem, the threshold scheme enables a secret key to be shared among p members in which each member holds a part of the secret key. In recent years, many works have been done on constructing (t + 1, l) threshold schemes using different algebraic aspects of the mathematics. In this paper, we discuss certain aspects of key sharing using Graph Theory and Strongly co-prime integers to present techniques which manage the sharing of keys equally amongst the users keeping the security of the system intact.
Keywords
Key Distribution, Chinese Remainder Theorem, Asymmetric Graph, Strongly co-prime integers, Graceful Labelling
A NOTE ON INDICES OF PRIMEPOWER AND SEMIPRIME DIVISOR FUNCTION GRAPH
A NOTE ON INDICES OF PRIMEPOWER AND SEMIPRIME DIVISOR FUNCTION GRAPH
S. Shanmugavelan, K. Thanga Rajeswari, C. Natarajan
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Abstract
The notion of using number theortic based graph seems to be one of the flourishing areas in Graph theory. One such concept is the divisor function graph GD(n) which is defined as: For any positive integer n ≥ 1 with r divisors d1,d2,d3,...,dr, divisor function graph GD(n) is a (V,E) graph with V as the set of all factors of n and E be defined in such a way that two vertices di and dj are adjacent if and only if either di | dj or dj | di, i ̸= j. In this paper, we analyze the operation sum of two divisor function graphs and investigate several indices exclusively for prime powers and for semi primes. Also, we derive a result for an independent function.
Keywords
Divisor function graph, Harmonic index, Zagreb index, Energy.
FUZZY ZERO DIVISOR GRAPH IN A COMMUTATIVE RING
FUZZY ZERO DIVISOR GRAPH IN A COMMUTATIVE RING
A. Kuppan, J. Ravi Sankar
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Abstract
Let R be a commutative ring and let Γ(Zn) be the zero divisor graph of a commutative ring R, whose vertices are non-zero zero divisors of Zn, and such that the two vertices u, v are adjacent if n divides uv. In this paper, we introduce the concept of fuzzy zero zivisor graph in a commutative ring and also discuss the some special cases of Γf (Z2p), Γf (Z3p), Γf (Z5p), Γf (Z7p) and Γf (Zpq). Throughout this paper we denote the Fuzzy Zero Divisor Graph(FZDG) by Γf (Zn ).
Keywords
Fuzzy graph, Zero divisor graph, Fuzzy zero divisor graph(FZDG).