Ambat Vijayakumar

Note: The {t}-property of some classes of graphs

In this note, the -properties of five classes of graphs are studied. We prove that the classes of cographs and clique perfect graphs without isolated vertices satisfy the -property and the -property, but do not satisfy the -property for t>=4. The -properties of the planar graphs and the perfect graphs are also studied. We obtain a necessary and sufficient condition for the trestled graph of index k to satisfy the -property.

Clique irreducibility of some iterative classes of graphs

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 Median Problem on k-Partite Graphs

Abstract In a connected graph G, the status of a vertex is the sum of the distances of that vertex to each of the other vertices in G. The subgraph induced by the vertices of minimum (maximum) status in G is called the median (anti-median) of G. The median problem of graphs is closely related to the optimization problems involving the placement of network servers, the core of the entire networks. Bipartite graphs play a significant role in designing very large interconnection networks. In this paper, we answer a problem on the structure of medians of bipartite graphs by showing that any bipartite graph is the median (or anti-median) of another bipartite graph. Also, with a different construction, we show that the similar results hold for k-partite graphs, k ≥ 3. In addition, we provide constructions to embed another graph as center in both bipartite and k-partite cases. Since any graph is a k-partite graph, for some k, these constructions can be applied in general

About triangles in a graph and its complement

Abstract The sum of number of triangles in a graph and its complement containing a vertex u is obtained and some results are deduced. A characterisation of F (G) is given, where G is a regular self-complementary graph and F (G) is the set of vertices in G each of which is contained in k(k−1) triangles. The definition of F (G) is extended to any simple graph G. A conjecture of Kotzig is also discussed.

Power Domination in Knödel Graphs and Hanoi Graphs

Abstract In this paper, we study the power domination problem in Knödel graphs WΔ,2ν and Hanoi graphs Hpn

Convex median and anti-median at prescribed distance

H_p^n

Strongly edge triangle regular graphs and a conjecture of Kotzig

. We determine the power domination number of W3,2ν and provide an upper bound for the power domination number of Wr+1,2r+1 for r ≥ 3. We also compute the k-power domination number and the k-propagation radius of Hp2

Some diameter notions in lexicographic product of graphs

H_p^2

On an edge partition and root graphs of some classes of line graphs

.

The Median Problem on Symmetric Bipartite Graphs

The status of a vertex v in a connected graph G is the sum of the distances between v and all the other vertices of G. The subgraph induced by the vertices of minimum (maximum) status in G is called median (anti-median) of G. Let