REMOVAL OF DEGENERACY IN THE TOPOLOGICAL INDICES OF COSPECTRAL GRAPHS USING EDGE WEIGHTED MOLECULAR GRAPHS

Two graphs having the same number of vertices connected in the same way are said to be isomorphic. Topological matrices representing such graphs will have the same eigen value spectrum. Hence isomorphic graphs are also known as co spectral or iso spectral graphs.  There are certain molecular graphs representing chemically different substances which are found to be co spectral graphs. Therefore the molecular descriptors calculated using topological matrices will not be able to distinguish between these molecules.  Under such circumstances, edge weighted topological matrices can be formulated so that, their eigen value spectrum will be unique. The edge weighted molecular graphs with bond length as the weight for the edges in adjacency and distance matrices distinguish between the molecules having similar molecular graphs. So different molecules having the same molecular graph will have the same topological indices, but the edge weighted adjacency and distance matrices have the ability to overcome this problem of degeneracy

Integrity and domination integrity of gear graphs

C.A. Barefoot, et. al. [4] introduced the concept of the integrity of a graph. It is an useful measure of vulnerability and it is defined as follows. I(G) = min{|S| + m(G − S) : S ⊂ V (G)}, where m(G − S) denotes the order of the largest component in G − S. Unlike the connectivity measures, integrity shows not only the difficulty to break down the network but also the damage that has been caused. A subset S of V (G) is said to be an I-set if I(G) = |S| + m(G − S). We introduced a new vulnerability parameter in [4],namely domination integrity of a graph G. It is a defined as DI(G) = min{|S| + m(G − S)}, where S is a dominating set of G and m(G − S) denotes the order of the largest component in G − S. K.S. Bagga,et. al. [2] gave a formula for I(K2 × Cn). In this paper, we give a correct formula for I(K2 × Cn). We find some results on the integrity and domination integrity of gear graphs.Publisher's Versio

Computational complexity of domination integrity in graphs

In a graph G, those dominating sets S which give minimum value for |S| + m(G−S), where m(G−S) denotes the maximum order of a component of G−S, are called dominating integrity sets of G (briefly called DI-sets of G). This concept combines two important aspects namely domination and integrity in graphs. In this paper, we Show that the decision problem domination integrity is NP-complete even when restricted to planar or chordal graphs.Publisher's Versio

Distance majorization sets in graphs

Let G = (V, E) be a simple graph. A subset D of V (G) is said to be a distance majorization set (or dm - set) if for every vertex u ∈ V − D, there exists a vertex v ∈ D such that d(u, v) ≥ deg(u) + deg(v). The minimum cardinality of a dm - set is called the distance majorization number of G (or dm - number of G) and is denoted by dm(G), Since the vertex set of G is a dm - set, the existence of a dm – set in any graph is guaranteed. In this paper, we find the dm - number of standard graphs like Kn, K1,n, Km,n, Cn, Pn, compute bounds on dm− number and dm- number of self complementary graphs and mycielskian of graphs.Publisher's Versio

Geodetic domination integrity in graphs

Reciprocal version of product degree distance of cactus graphs Let G be a simple graph. A subset S ⊆ V (G) is a said to be a geodetic set if every vertex u /∈ S lies on a shortest path between two vertices from S. The minimum cardinality of such a set S is the geodetic number g(G) of G. A subset D ⊆ V (G) is a dominating set of G if every vertex u /∈ D has at least one neighbor in D. The domination number γ(G) is the minimum cardinality of a dominating set of G. A subset is said to be a geodetic dominating set of G if it is both a geodetic and a dominating set. The geodetic domination number γg(G) is the minimum cardinality among all geodetic dominating sets in G. The geodetic domination integrity of a graph G is defined by DIg(G) = min{|S| + m(G − S) : S is a geodetic dominating set of G}, where m(G − S) denotes the order of the largest component in G−S. In this paper, we study the concepts of geodetic dominating integrity of some families of graphs and derive some bounds for the geodetic domination integrity. Also we obtain geodetic domination integrity of some cartesian product of graphs.Publisher's Versio

Interval- valued fermatean neutrosophic graphs

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Tight just excellent graphs

A graph G is χ-excellent if for every vertex v, there exists a chromatic partition π such that {v} ∈ π.A graph G is just χ-excellent if every vertex appears as a singleton in exactly one χ-partition. In this paper, a special type of just χ-excellence namely tight just χ-excellence is defined and studied.Publisher's Versio

On zagreb indices of double vertex graphs

Let G = (V, E) be a graph with at least 2 vertices, then the double vertex graph U₂(G) is the graph whose vertex set consists of all 2-subsets of V such that two distinct vertices {x, y} and {u, v} are adjacent if and only if |{x, y} ∩ {u, v}| = 1 and if x = u, then y and v are adjacent in G. Similarly, the complete double vertex graph, denoted by CU₂(G), has vertex set consists of all unordered pairs of elements of V and two distinct vertices {x, y} and {u, v} are adjacent if and only if |{x, y} ∩ {u, v}| = 1 and if x = u, then y and v are adjacent in G. In this work, we compute the zagreb indices of double vertex and complete double vertex graphs.The authors would like to thank the Management and Principal, SSN College of Engineering, Chennai, India.Publisher's Versio

Response of bacterioplankton to iron fertilization of theSouthern Ocean, Antarctica

Ocean iron fertilization is an approach to increase CO2 sequestration. The Indo-German iron fertilization experiment LOHAFEX was carried out in the Southern Ocean surrounding Antarctica in 2009 to monitor changes in bacterial community structure following iron fertilization-induced phytoplankton bloom of the seawater from different depths. 16S rRNA gene libraries were constructed using metagenomic DNA from seawater prior to and after iron fertilization and the clones were sequenced for identification of the major bacterial groups present and for phylogenetic analyses. A total of 4439 clones of 16S rRNA genes from ten 16S rRNA gene libraries were sequenced. More than 97.35% of the sequences represented four bacterial lineages i.e. Alphaproteobacteria, Gammaproteobacteria, Bacteroidetes and Firmicutes and confirmed their role in scavenging of phytoplankton blooms induced following iron fertilization. The present study demonstrates the response of Firmicutes due to Iron fertilization which was not observed in previous southern ocean Iron fertilization studies. In addition, this study identifies three unique phylogenetic clusters LOHAFEX Cluster 1 (affiliated to Bacteroidetes), 2 and 3 (affiliated to Firmicutes) which were not detected in any of the earlier studies on iron fertilization. The relative abundance of these clusters in response to iron fertilization was different. The increase in abundance of LOHAFEX Cluster 2 and Papillibacter sp. another dominant Firmicutes may imply a role in phytoplankton degradation. Disappearance of LOHAFEX Cluster 3 and other bacterial genera after iron fertilization may imply conditions not conducive for their survival. It is hypothesised that heterotrophic bacterial abundance in the Southern Ocean would depend on their ability to utilise algal exudates, decaying algal biomass and other nutrients thus resulting in a dynamic bacterial succession of distinct genera