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.

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

Department of Mathematics, Cochin University of Science and Technology

Bioactivity Profile of Polyunsaturated Fattyacid extracts from Sardinella longiceps and Sardinella fimbriata – A comparative study

The oceans have proved to be an interminable source of new and effective drugs. Innumerable studies have proved that specific compounds isolated from marine organisms have great nutritional and pharmaceutical value. Polyunsaturated fattyacids (PUFA) in general are known for their dietary benefits in preventing and curing several critical ailments including Coronary heart disease (CHD) and cancers of various kinds. Eicosapentaenoic Acid (EPA) and Docosahexaenoic Acid (DHA) are two PUFA which are entirely marine in origin – and small Clupeoid fishes like sardines are known to be excellent sources of these two compounds. In this study, we selected two widely available Sardine species in the west coast, Sardinella longiceps and Sardinella fimbriata, for a comparative analysis of their bioactive properties. Both these sardines are known to be rich in EPA and DHA, however considerable seasonal variation in its PUFA content was expected and these variations studied. An extraction procedure to isolate PUFA at high purity levels was identified and the extracts obtained thus were studied for anti-bacterial, anti-diabetic and anti-cancerous properties.Samples of both the sardines were collected from landing centre, measured and their gut content analysed in four different months of the year – viz. June, September, December and March. The fish samples were analyzed for fattyacid using FAME method using gas chromatography to identify the full range of fattyacids and their respective concentration in each of the samples. The fattyacids were expressed in mg/g meat and later converted to percentage values against total fatty acids and total PUFA content. Fattyacids during winter season (Dec-Mar) were found to be generally higher than spawning season (June-Sept). PUFA dominated the profiles of both species and average PUFA content was also higher during winter. However, it was found that S. longiceps had proportionately higher EPA as compared to S. fimbriata which was DHA rich. Percentage of EPA and DHA also varied across months for both species – the spawning season seemed to show higher EPA content in S. longiceps and higher DHA content in S. fimbriata. Gut content analysis indicate that adult S. fimbriata is partial to zooplanktons which are DHA rich while adult S. longiceps feed mainly on EPA rich phytoplankton. Juveniles of both species, found mainly in winter, had a gut content showing more mixed diet. This difference in the feeding pattern reflect clearly in their PUFA profile – adult S. longiceps, which dominate the catch during the spawn season, feeding mostly on phytoplankton is concentrated with EPA while the juveniles which are found mostly in the winter season has slightly less EPA proportion as compared to adults. The same is true for S. fimbriata adults that are caught mostly in the spawning season; being rich in DHA as they feed mainly on zooplankton while the juveniles caught during winter season has a relatively lower concentration of DHA in their total PUFA.Various extraction procedures are known to obtain PUFA from fish oil. However, most of them do not give high purity and do not use materials indicated as safe. PUFA extracts have to be edible and should not have harmful substances for applying on mice and human subjects. Some PUFA extraction procedures, though pure and non-toxic, might induce cis-trans conversions during the extraction process. This conversion destroys the benefits of PUFA and at times is harmful to human body. A method free from these limitations has been standardized for this study. Gas Chromatography was performed on the extracts thus made to ensure that it is substantially pure. EPA: DHA ratios for both samples were derived - for S. longiceps this ratio was 3:2, while it was 3:8 for S. fimbriata.Eight common strains of gram positive and gram negative bacterial strains were subjected to the PUFA extracts from both species dissolved in acetone solution using Agar Well Diffusion method. The activity was studied against an acetone control. At the end of incubation period, zones of inhibition were measured to estimate the activity. Minimum inhibitory concentration for each of the active combinations was calculated by keeping p < 0.01 as significant. Four of the bacteria including multi-resistant Staphylococcus aureus were shown to be inhibited by the fish extracts. It was also found that the extracts from S. fimbriata were better than the one from S. longiceps in annihilating harmful bacteria.Four groups of mice subjects were studied to evaluate the antidiabetic properties of the PUFA extracts. Three groups were induced diabetes by administration of alloxan tetra hydrate. One group without diabetes was kept as control and another with diabetes was kept as diabetic control. For two diabetic groups, a prescribed amount of fish extracts were fed from each of the extracts. The biochemical parameters like serum glucose, total cholesterol, LDL & HDL cholesterol, triglycerides, urea and creatinine were sampled from all four groups at regular intervals of 7 days for a period of 28 days. It was found that groups fed with fish extracts had marked improvement in the levels of total LDL & HDL cholesterol, triglycerides and creatinine. Groups fed with extracts from S. fimbriata seem to have fared better as compared to S. longiceps. However, both groups did not show any marked improvement in blood glucose levels or levels of urea.Cell lines of MCF-7 (Breast Cancer) and DU-145 (Prostate Cancer) were used to analyse the cytotoxicity of the PUFA extracts. Both cell lines were subjected to MTT Assay and later the plates were read using an ELISA reader at a wavelength of 570nm. It was found that both extracts had significant cytotoxic effects against both cell lines and a peak cytotoxicity of 85-90% was apparent. IC50 values were calculated from the graphs and it was found that S. longiceps extracts had a slightly lower IC50 value indicating that it is toxic even at a lower concentration as compared to extracts from S. fimbriata.This study summarizes the bioactivity profile of PUFA extracts and provides recommendation for dietary intake; fish based nutritional industry and indigenous pharmaceutical industry. Possible future directions of this study are also elaborated.

Clique Irreducibility and Clique Vertex Irreducibility of Graphs

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.

Studies on the Spectrum and the Energy of Graphs

Department of Mathematics, Cochin University of Science and Technology

On infinite graphs and related matrices

This thesis Entitled On Infinite graphs and related matrices.ln the last two decades (iraph theory has captured wide attraction as a Mathematical model for any system involving a binary relation. The theory is intimately related to many other branches of Mathematics including Matrix Theory Group theory. Probability. Topology and Combinatorics . and has applications in many other disciplines..Any sort of study on infinite graphs naturally involves an attempt to extend the well known results on the much familiar finite graphs. A graph is completely determined by either its adjacencies or its incidences. A matrix can convey this information completely. This makes a proper labelling of the vertices. edges and any other elements considered, an inevitable process. Many types of labelling of finite graphs as Cordial labelling, Egyptian labelling, Arithmetic labeling and Magical labelling are available in the literature. The number of matrices associated with a finite graph are too many For a study ofthis type to be exhaustive. A large number of theorems have been established by various authors for finite matrices. The extension of these results to infinite matrices associated with infinite graphs is neither obvious nor always possible due to convergence problems. In this thesis our attempt is to obtain theorems of a similar nature on infinite graphs and infinite matrices. We consider the three most commonly used matrices or operators, namely, the adjacency matrix

‘Studies on some topics in product graphs’

For routing problems in interconnection networks it is important to find the shortest containers between any two vertices, since the w-wide diameter gives the maximum communication delay when there are up to w−1 faulty nodes in a network modeled by a graph. The concept of ‘wide diameter’ was introduced by Hsu [41] to unify the concepts of diameter and The concept of ‘domination’ has attracted interest due to its wide applications in many real world situations [38]. A connected dominating set serves as a virtual backbone of a network and it is a set of vertices that helps in routing. In this thesis, we make an earnest attempt to study some of these notions in graph products. This include, the diameter variability, the diameter vulnerability, the component factors and the domination criticality.connectivity

Computing median and antimedian sets in median graphs

The median (antimedian) set of a profile π = (u1, . . . , uk) of vertices of a graphG is the set of vertices x that minimize (maximize) the remoteness i d(x,ui ). Two algorithms for median graphs G of complexity O(nidim(G)) are designed, where n is the order and idim(G) the isometric dimension of G. The first algorithm computes median sets of profiles and will be in practice often faster than the other algorithm which in addition computes antimedian sets and remoteness functions and works in all partial cubes

On the remoteness function in median graphs

A profile on a graph G is any nonempty multiset whose elements are vertices from G. The corresponding remoteness function associates to each vertex x 2 V.G/ the sum of distances from x to the vertices in the profile. Starting from some nice and useful properties of the remoteness function in hypercubes, the remoteness function is studied in arbitrary median graphs with respect to their isometric embeddings in hypercubes. In particular, a relation between the vertices in a median graph G whose remoteness function is maximum (antimedian set of G) with the antimedian set of the host hypercube is found. While for odd profiles the antimedian set is an independent set that lies in the strict boundary of a median graph, there exist median graphs in which special even profiles yield a constant remoteness function. We characterize such median graphs in two ways: as the graphs whose periphery transversal number is 2, and as the graphs with the geodetic number equal to 2. Finally, we present an algorithm that, given a graph G on n vertices and m edges, decides in O.mlog n/ time whether G is a median graph with geodetic number 2

Strongly distance-balanced graphs and graph products

A graph G is strongly distance-balanced if for every edge uv of G and every i 0 the number of vertices x with d.x; u/ D d.x; v/ 1 D i equals the number of vertices y with d.y; v/ D d.y; u/ 1 D i. It is proved that the strong product of graphs is strongly distance-balanced if and only if both factors are strongly distance-balanced. It is also proved that connected components of the direct product of two bipartite graphs are strongly distancebalanced if and only if both factors are strongly distance-balanced. Additionally, a new characterization of distance-balanced graphs and an algorithm of time complexity O.mn/ for their recognition, wheremis the number of edges and n the number of vertices of the graph in question, are given

Simultaneous Embeddings Of Graphs As Median And Antimedian Subgraphs

The distance DG(v) of a vertex v in an undirected graph G is the sum of the distances between v and all other vertices of G. The set of vertices in G with maximum (minimum) distance is the antimedian (median) set of a graph G. It is proved that for arbitrary graphs G and J and a positive integer r 2, there exists a connected graph H such that G is the antimedian and J the median subgraphs of H, respectively, and that dH(G, J) = r. When both G and J are connected, G and J can in addition be made convex subgraphs of H.

The Plurality Strategy on Graphs

The Majority Strategy for finding medians of a set of clients on a graph can be relaxed in the following way: if we are at v, then we move to a neighbor w if there are at least as many clients closer to w than to v (thus ignoring the clients at equal distance from v and w). The graphs on which this Plurality Strategy always finds the set of all medians are precisely those for which the set of medians induces always a connected subgraph