Studies on Fuzzy Graphs

In this thesis an attempt to develop the properties of basic concepts in fuzzy graphs such as fuzzy bridges, fuzzy cutnodes, fuzzy trees and blocks in fuzzy graphs have been made. The notion of complement of a fuzzy graph is modified and some of its properties are studied. Since the notion of complement has just been initiated, several properties of G and G available for crisp graphs can be studied for fuzzy graphs also. Mainly focused on fuzzy trees defined by Rosenfeld in [10] , several other types of fuzzy trees are defined depending on the acyclicity level of a fuzzy graph. It is observed that there are selfcentered fuzzy trees. Some operations on fuzzy graphs and prove that complement of the union two fuzzy graphs is the join of their complements and complement of the join of two fuzzy graphs is union of their complements. The study of fuzzy graphs made in this thesis is far from being complete. The wide ranging applications of graph theory and the interdisciplinary nature of fuzzy set theory, if properly blended together could pave a way for a substantial growth of fuzzy graph theory.

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.

On Distance Energy of Graphs

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.

A Note on Energy of Some Graphs

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.

Equienergetic self-complementary graphs

In this paper equienergetic self-complementary graphs on p vertices for every p = 4k; k ¸ 2 and p = 24t + 1; t ¸ 3 are constructed

Some classes of Turker equivalent graphs

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 (t) property of some classes of graphs

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

SOME NEW INTEGRAL GRAPHS

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.

ON DISTANCE ENERGY OF GRAPHS

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.

GALLAI AND ANTI-GALLAI GRAPHS OF A GRAPH

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.

Antimedian graphs

Antimedian graphs are introduced as the graphs in which for every triple of vertices there exists a unique vertex x that maximizes the sum of the distances from x to the vertices of the triple. The Cartesian product of graphs is antimedian if and only if its factors are antimedian. It is proved that multiplying a non-antimedian vertex in an antimedian graph yields a larger antimedian graph. Thin even belts are introduced and proved to be antimedian. A characterization of antimedian trees is given that leads to a linear recognition algorithm.

Energies of some non-regular graphs

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.

Studies on ocean surface layer responses to atmospheric forcing in the North Indian Ocean

The present study on upper ocean responses to atmospheric forcing (associated with cyclone passage) in North Indian Ocean revealed significant variability between AS and BoB. The analysis of cyclone frequency during 1947 to 2006 exhibited lesser frequency of cyclones in AS than that of BoB. The analysis also revealed significant reduction in cyclone frequency after the year 1976 with substantial reduction during monsoon season. The long term SST data at selected points in AS and BoB could not reveal any relation with reduction in cyclone frequency. However the SLP at same locations exhibited considerable increase during mid 1970’s, which could have contributed to the observed reduction in cyclone frequency after the year 1976.The response in waves during cyclone passage exhibited significant asymmetry on either side of the track in AS and BoB and the response is observed at 100’s of kilometers away from the track. The significant clockwise rotation in wave direction is observed on the right side of the track starting from near the track to far away locations, which existed for a longer duration. However, the anticlockwise rotation in wave direction is observed over a shorter distance on the left side of the track and dissipated immediately.Inertial oscillation is observed in surface current and in the mixed layer temperature associated with cyclone passage, which revealed the role of relative location(s) on either side of the track. The inertial peak closer to the local inertial period indicates maximum transfer of energy during the cyclone passage in both AS and BoB. The absence of strong inertial oscillation even with clockwise rotation in surface current and wind indicates the dominant role of duration of strong wind in generating inertial oscillation.The oceanic response associated with cyclone passage reveal the variable response(s) which depends on cyclone intensity, the proximity to track and cyclone translation speed. It is observed that resonance with wind generates higher response in surface current, wave and SST on the right side of the track and it lasts for a longer duration. The maximum oceanic response is observed at a few kilometers away on right side of the track. However lesser rightward bias in the location of maximum cooling is observed for cyclones with low cyclone translation speed. The response on the left side of the track is less and is limited over a shorter distance and dissipates immediately. It is observed that the ocean response, in general, increases with intensity of cyclones. However the differential cooling produced by the same intensity cyclones in AS and in BoB indicates the dominant role of low cyclone translation speed in oceanic response.The surface cooling exhibited strikingly differential responses between AS and BoB. The TMI-SST and buoy observations exhibited significant cooling for a longer duration in AS compared to that of BoB. The spatial extent of cooling is also much higher in AS than that of BoB. The wide spread cooling associated with cyclone passage in AS indicates the dominant role of thermal structure in oceanic response in AS than that of BoB.

Studies on some generalizations of line graph and the power domination problem in graphs

Department of Mathematics, Cochin University of Science and Technology