Equitable and Non-Equitable Zagreb Indices of Graphs

In this paper, we present exact formula for the weighted PI index of corona product of two connected graphs in terms of other graph invariants including the PI index, first Zagreb index and second Zagreb index. Then, we apply our result to compute the weighted PI indices of t-fold bristled graph, bottleneck graph, sunlet graph, star graph, fan graph, wheel graph and some classes of bridge graphs.

The International J.Mathematical Combinatorics is a fully refereed international journal, sponsored by the MADIS of Chinese Academy of Sci ences and published in USA quarterly comprising 110-160 pages approx. per volume, which publishes original research papers and survey articles in all aspects of Smarandachemulti-spaces, Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topology and their applications to other sciences.

Let G = (V, E) be a connected graph. The distance eccentricity neighborhood of u ∈ V (G) denoted by NDe(u) is defined as NDe(u) = {v ∈ V (G) : d(u, v) = e(u)}, where e(u) is the eccentricity of u. The cardinality of NDe(u) is called the distance eccentricity degree of the vertex u in G and denoted by deg De (u). In this paper, we introduce the first and second distance eccentricity Zagreb indices of a connected graph G as the sum of the squares of the distance eccentricity degrees of the vertices, and the sum of the products of the distance eccentricity degrees of pairs of adjacent vertices, respectively. Exact values for some families of graphs and graph operations are obtained.

The reformulated first Zagreb index is the edge version of first Zagreb index of chemical graph theory. The aim of this paper is to obtain an expression for the reformulated first Zagreb index of the some class of graphs such as Tadpole graph, Wheel graph, Ladder graph. Further we also obtain the reformulated first Zagreb index of the line graph, subdivision graph and line graph of subdivision graph for class of graphs.

2018

Applications of Smarandache multi-spaces to theoretical physics; Applications of Combinatorics to mathematics and theoretical physics; Mathematical theory on gravitational fields; Mathematical theory on parallel universes; Other applications of Smarandache multi-space and combinatorics.

The Mathematical Combinatorics (International Book Series) is a fully refereed international book series with ISBN number on each issue, sponsored by the MADIS of Chinese Academy of Sciences and published in USA quarterly comprising 100-150 pages approx. per volume, which publishes original research papers and survey articles in all aspects of Smarandache multi-spaces, Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topology and their applications to other sciences.

2010

It was conjectured that for each simple graph G = (V , E) with n = |V (G)| vertices and m = |E(G)| edges, it holds M 2 (G)/m ≥ M 1 (G)/n, where M 1 and M 2 are the first and second Zagreb indices. Hansen and Vukičević proved that it is true for all chemical graphs and does not hold in general. Also the conjecture was proved for all trees, unicyclic graphs, and all bicyclic graphs except one class. In this paper, we show that for every positive integer k, there exists a connected graph such that m − n = k and the conjecture does not hold. Moreover, by introducing some transformations, we show that M 2 /(m − 1) > M 1 /n for all bicyclic graphs and it does not hold for general graphs. Using these transformations we give new and shorter proofs of some known results.

The International J.Mathematical Combinatorics (ISSN 1937-1055) is a fully refereed international journal, sponsored by the MADIS of Chinese Academy of Sciences and published in USA quarterly comprising 100-150 pages approx. per volume, which publishes original research papers and survey articles in all aspects of Smarandachemulti-spaces, Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topology and their applications to other sciences.

The International J.Mathematical Combinatorics (ISSN 1937-1055) is a fully refereed international journal, sponsored by the MADIS of Chinese Academy of Sciences and published in USA quarterly comprising 100-150 pages approx. per volume, which publishes original research papers and survey articles in all aspects of Smarandache multi-spaces, Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topology and their applications to other sciences.

Abstract Let G be a non-abelian group and let Z (G) be the center of G. The noncommuting graph of G, Γ (G), is a graph with vertex set G\ Z (G) and two distinct vertices x and y are adjacent if and only if xy= yx. In this paper the Hyper-Wiener, Schultz, Gutman, eccentric connectivity and Zagreb group indices of this graph are computed. Keywords: Non-commuting graph, AC-group, Wiener index, Hyper-Wiener index, Schultz index, Zagreb index.