A graphs G is clique irreducible if every clique in G of size at least two,has an edge which does not lie in any other clique of G and is clique reducible if it is not clique irreducible. A graph G is clique vertex irreducible if every clique in G has a vertex which does not lie in any other clique of G and clique vertex reducible if it is not clique vertex irreducible. The clique vertex irreducibility and clique irreducibility of graphs which are non-complete extended p-sums (NEPS) of two graphs are studied. We prove that if G(c) has at least two non-trivial components then G is clique vertex reducible and if it has at least three non-trivial components then G is clique reducible. The cographs and the distance hereditary graphs which are clique vertex irreducible and clique irreducible are also recursively characterized.
Aparna,Lakshmanan S; Vijayakumar,A(Department of Mathematics, 2008)
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Abstract:
In this paper, two notions, the clique irreducibility and clique vertex
irreducibility are discussed. A graph G is clique irreducible if every
clique in G of size at least two, has an edge which does not lie in any
other clique of G and it is clique vertex irreducible if every clique in G
has a vertex which does not lie in any other clique of G. It is proved
that L(G) is clique irreducible if and only if every triangle in G has a
vertex of degree two. The conditions for the iterations of line graph,
the Gallai graphs, the anti-Gallai graphs and its iterations to be clique
irreducible and clique vertex irreducible are also obtained.
The concept of convex extendability is introduced to answer the problem of finding the smallest
distance convex simple graph containing a given tree. A problem of similar type with respect
to minimal path convexity is also discussed.
Abstract. The edge C4 graph E4(G) of a graph G has all the edges of Gas its
vertices, two vertices in E4(G) are adjacent if their corresponding edges in G are
either incident or are opposite edges of some C4. In this paper, characterizations
for E4(G)
being connected, complete, bipartite, tree etc are given. We have
also proved that E4(G) has no forbidden subgraph characterization. Some
dynamical behaviour such as convergence, mortality and touching number are also studied
Indulal,G; Vijayakumar,A(Springer, October , 2007)
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Abstract:
The energy of a graph G is the sum of the absolute values of its eigenvalues. In this
paper, we study the energies of some classes of non-regular graphs. Also the spectrum
of some non-regular graphs and their complements are discussed.
Lakshmanan,Aparna; Rao, S B; Vijayakumar,A(February 4, 2010)
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Abstract:
Abstract. The paper deals with graph operators-the Gallai graphs and the anti-Gallai
graphs. We prove the existence of a finite family of forbidden subgraphs for the Gallai graphs
and the anti-Gallai graphs to be H-free for any finite graph H. The case of complement
reducible graphs-cographs is discussed in detail. Some relations between the chromatic
number, the radius and the diameter of a graph and its Gallai and anti-Gallai graphs are
also obtained.
Indulal,G; Vijayakumar,A(Department of Mathematics, 2008)
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Abstract:
Eigenvalue of a graph is the eigenvalue of its adjacency matrix. The energy of a graph is the
sum of the absolute values of its eigenvalues. In this note we obtain analytic expressions for the
energy of two classes of regular graphs.
Aparna,Lakshmanan S; Vijayakumar,Ambat(Department of Mathematics, 2000)
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Abstract:
In this paper, we study the domination number, the global dom
ination number, the cographic domination number, the global co
graphic domination number and the independent domination number
of all the graph products which are non-complete extended p-sums
(NEPS) of two graphs.
Gopalapillai,Indulal; Ivan,Gutman; Vijayakumar,Ambat(Department of Mathematics, August 25, 2007)
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Abstract:
The D-eigenvalues of a graph G are the eigenvalues of its distance matrix D, and the
D-energy ED(G) is the sum of the absolute values of its D-eigenvalues. Two graphs are
said to be D-equienergetic if they have the same D-energy. In this note we obtain bounds
for the distance spectral radius and D-energy of graphs of diameter 2. Pairs of equiregular
D-equienergetic graphs of diameter 2, on p = 3t + 1 vertices are also constructed.
The D-eigenvalues of a graph G are the eigenvalues of its distance matrix D, and the
D-energy ED(G) is the sum of the absolute values of its D-eigenvalues. Two graphs are
said to be D-equienergetic if they have the same D-energy. In this note we obtain bounds
for the distance spectral radius and D-energy of graphs of diameter 2. Pairs of equiregular
D-equienergetic graphs of diameter 2, on p = 3t + 1 vertices are also constructed.
this paper, the median and the antimedian of cographs are
discussed. It is shown that if G, and G2 are any two cographs, then there is a
cograph that is both Eulerian and Hamiltonian having Gl as its median and G2
as its antimedian. Moreover, the connected planar and outer planar cographs
are characterized and the median and antimedian graphs of connected, planar
cographs are listed.
We define a new graph operator called the P3 intersection graph,
P3(G)- the intersection graph of all induced 3-paths in G. A characterization
of graphs G for which P-3 (G) is bipartite is given . Forbidden
subgraph characterization for P3 (G) having properties of being
chordal , H-free, complete are also obtained . For integers a and b
with a > 1 and b > a - 1, it is shown that there exists a graph G
such that X(G) = a, X(P3( G)) = b, where X is the chromatic number
of G. For the domination number -y(G), we construct graphs G such
that -y(G) = a and -y (P3(G)) = b for any two positive numbers a > 1
and b. Similar construction for the independence number and radius,
diameter relations are also discussed.
Indulal,G; Vijayakumar,Ambat(Department of Mathematics, 2002)
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Abstract:
Two graphs G and H are Turker equivalent if they have the same set of Turker angles.
In this paper some Turker equivalent family of graphs are obtained.
The eigenvalue of a graph is the eigenvalue of its adjacency matrix . A graph
G is integral if all of its cigenvalues are integers. In this paper some new
classes of integral graphs are constructed.
In this note,the (t) properties of five class are studied. We proved that the classes of cographs and clique perfect graphs without isolated vertices satisfy the (2) property and the (3) property, but do not satisfy the (t) property for tis greater than equal to 4. The (t) properties of the planar graphs and the perfect graphss are also studied . we obtain a necessary and suffieient conditions for the trestled graph of index K to satisfy the (2) property