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Two new characterizations of consistent marked graph
E. Sampathkumar 장전수학회 2011 Advanced Studies in Contemporary Mathematics Vol.21 No.4
A consistent marked graph is a graph in which every vertex is signed + or −, and where every cycle has an even number of negative vertices. Theorem. For a consistent marked graph G, the following statements are equivalent 1) G is consistent. 2) There exists a partition {E_1,E-2} of the edge set of each cycle C in G such that every negative vertex on C, appears as an end vertex of an edge in each of the sets E_1 and E_2, and no positive vertex on C has this property. 3) The edge set of each cycle C in G can be partitioned into two sets {E_1,E_2} such that any two adjacent edges in C belong to different sets in {E_1,E_2} if, and only if, they have a common negative vertex.
E. Sampathkumar,P. S. K. Reddy,M. S. Subramanya 장전수학회 2008 Proceedings of the Jangjeon mathematical society Vol.11 No.1
An n-tuple (a₁, a₂, ..., an) is symmetric, if ak = an−k+₁, 1 ≤ k ≤ n. A symmetric n-sigraph (symmetric n-marked graph) is an ordered pair Sn = (G, δ) (Sn = (G, μ)), where G = (V,E) is a graph called the underlying graph of Sn and δ : E → Hn (μ : V → Hn) is a function. Analogous to the concept of Jump sigraph of a sigraph, we define Jump symmetric n-sigraph of a symmetric n-sigraph. Introducing two notions of balance in symmetric n-sigraphs and some notions of complements we characterize the symmetric n-sigraphs which are switching equivalent to their Jump symmetric n-sigraphs.
ON CERTAIN ENERGIES OF A ONE-POINT UNION OF COMPLETE GRAPHS Kn Κn
E. Sampathkumar,R. BHARATI,K. SATHISH,SUDEEP STEPHEN 장전수학회 2018 Proceedings of the Jangjeon mathematical society Vol.21 No.1
The eigenvalues of a graph G are the eigenvalues of its adjacency matrix. The energy of the graph is defined as the sum of the absolute values of all its eigenvalues. In this paper we compute different energies of a one-point union of m copies of complete graphs on n vertices.
Matrix Representation of Disemigraphs
E. Sampathkumar,L. Pushpalatha 장전수학회 2013 Advanced Studies in Contemporary Mathematics Vol.23 No.1
A disemigraph can be uniquely represented by a matrix and a characterization of such a matrix is obtained.
E. Sampathkumar,S. V. Roopa,K. A. Vidya,M. A. Sriraj 장전수학회 2015 Proceedings of the Jangjeon mathematical society Vol.18 No.4
Let G = (V,E) be a graph. Let V1, V2, . . . , Vk be non-empty disjoint subsets of V such that union equal to V . Then {V1, V2, . . . , Vk} is called partition of vertex set V . Using this partition the graph G can be uniquely represented by a matrix called L-matrix Pk(G), whose entries belong to the set {2, 1, 0,−1} and defined as follows: aij = 8>>< >>: 2 if vi and vj are adjacent within the partition Vi, −1 if vi and vj are non-adjacent within the partition Vi, 1 if vi and vj are adjacent between the partition Vi and Vj for i 6= j, 0 otherwise. The eigenvalues of this matrix are called k-partition eigenvalues of G. The k-partition energy EPk (G) is defined as the sum of the absolute values of kpartition eigenvalues of G. We determine partition energy of some known graphs and also obtain bounds for EPk (G).
Partition energy of complete product of circulant graphs and some new class of graphs
E. Sampathkumar,S. V. Roopa,K. A. Vidya,M. A. Sriraj 장전수학회 2018 Advanced Studies in Contemporary Mathematics Vol.28 No.2
Let G = (V,E) be a graph and Pk = {V1, V2, ..., Vk} be a partition of V . The L-matrix with respect to a partition Pk of the vertex set V of graph G of order n is the unique square symmetric matrix Pk(G) = [aij ] with zero diagonal, whose entries aij with i 6≠ j are defined as follows: (i) If vi, vj ∈ Vr, then aij = 2 or −1 according as vivj is an edge or not. (ii) If vi ∈ Vr and vj ∈ Vs for r 6≠s, then aij = 1 or 0 according as vivj is an edge or not. For all Vi and Vj in Pk, i 6≠j remove the edges between vertices of Vi and Vj and add the edges between the vertices of Vi and Vj which are not in G, the resulting graph is called k-complement of G and is denoted by (G)k. For each set Vr in Pk, remove the edges of G joining the vertices within Vr and add the edges of G (complement of G) joining the vertices of Vr, the graph obtained is called k(i)-complement and is denoted by (G)k(i). The k-partition energy of a graph G with respect to partition Pk is denoted by EPk (G) and is defined as the sum of the absolute values of k-partition eigenvalues of Pk(G). In this paper we construct some graphs such that the graph and its 2-complement are equienergetic with respect to a given partition. We also determine partition energy of complete product of m copies of a circulant graph G and its subgraph, their k-complement and k(i)-complement.
Matrix representation of semigraphs
E. Sampathkumar,L. Pushpalatha 장전수학회 2007 Advanced Studies in Contemporary Mathematics Vol.14 No.1
This paper deals with representations of a semigraph by matrices. The adjacency matrix, the incidence matrix, the consecutive adjacency matrix and the 3-matrix of a semigraph are defined. The incidence matrix, together with the consecutive adjacency matrix, determines a semigraph uniquely. Also, the 3-matrix of a semigraph G determines G uniquely.
(3,d)-sigraph and its applications
E. Sampathkumar,P. S. K. Reddy,M. S. Subramanya 장전수학회 2008 Advanced Studies in Contemporary Mathematics Vol.17 No.1
LetG = (V;E) be a graph and x = uv be an edge in G. By directional labeling (or d-labeling) of an edge x = uv of G by an ordered triple(a1;a2;a3), we mean a labeling of x such that we consider the label on x as (a1;a2;a3) in the direction from u to v, and the label on x as (a3;a2;a1)in the direction from v to u. In this paper, we study graphs in which every edge is d-labeled by a triple (a1;a2;a3), where ak 2 f+ ;g , for 1 k 3,called (3, d)-sigraphs. Giving some motivation to study such graphs, we obtain some results by introducing some notions of balance and special types of complements.
Chandrashekar Adiga,E. Sampathkumar,M. A. Sriraj,Shrikanth A S 장전수학회 2013 Proceedings of the Jangjeon mathematical society Vol.16 No.3
In this paper, we introduce the concept of color energy of a graph, Ec(G) andcompute the color energy Ex(G) of few families of graphs with minimum number ofcolors. It depends on the underlying graph and colors on its vertices. We establishan upper bound and a lower bound for color energy. Also we introduce the conceptof complement of a colored graph and compute energies of complement of coloredgraphs of few families of graphs.