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.
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.
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(Springer, October , 2007)
[+]
[-]
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)
[+]
[-]
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.
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.
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.