1,084 research outputs found

    Extremal structures of graphs with given connectivity or number of pendant vertices

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    For a graph GG, the first multiplicative Zagreb index ∏1(G)\prod_1(G) is the product of squares of vertex degrees, and the second multiplicative Zagreb index ∏2(G)\prod_2(G) is the product of products of degrees of pairs of adjacent vertices. In this paper, we explore graphs with extremal Π1(G)\Pi_{1}(G) and Π2(G)\Pi_{2}(G) in terms of (edge) connectivity and pendant vertices. The corresponding extremal graphs are characterized with given connectivity at most kk and pp pendant vertices. In addition, the maximum and minimum values of ∏1(G)\prod_1(G) and ∏2(G)\prod_2(G) are provided. Our results extend and enrich some known conclusions.Comment: arXiv admin note: substantial text overlap with arXiv:1704.0694

    Simple Alcohols with the Lowest Normal Boiling Point Using Topological Indices

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    We find simple saturated alcohols with the given number of carbon atoms and the minimal normal boiling point. The boiling point is predicted with a weighted sum of the generalized first Zagreb index, the second Zagreb index, the Wiener index for vertex-weighted graphs, and a simple index caring for the degree of a carbon atom being incident to the hydroxyl group. To find extremal alcohol molecules we characterize chemical trees of order nn, which minimize the sum of the second Zagreb index and the generalized first Zagreb index, and also build chemical trees, which minimize the Wiener index over all chemical trees with given vertex weights.Comment: 22 pages, 5 figures, accepted in 2014 by MATCH Commun. Math. Comput. Che

    On the sharp lower bounds of Zagreb indices of graphs with given number of cut vertices

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    The first Zagreb index of a graph GG is the sum of the square of every vertex degree, while the second Zagreb index is the sum of the product of vertex degrees of each edge over all edges. In our work, we solve an open question about Zagreb indices of graphs with given number of cut vertices. The sharp lower bounds are obtained for these indices of graphs in Vn,k\mathbb{V}_{n,k}, where Vn,k\mathbb{V}_{n, k} denotes the set of all nn-vertex graphs with kk cut vertices and at least one cycle. As consequences, those graphs with the smallest Zagreb indices are characterized.Comment: Accepted by Journal of Mathematical Analysis and Application

    Efficient computation of trees with minimal atom-bond connectivity index

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    The {\em atom-bond connectivity (ABC) index} is one of the recently most investigated degree-based molecular structure descriptors, that have applications in chemistry. For a graph GG, the ABC index is defined as βˆ‘uv∈E(G)(d(u)+d(v)βˆ’2)d(u)d(v)\sum_{uv\in E(G)}\sqrt{\frac{(d(u) +d(v)-2)}{d(u)d(v)}}, where d(u)d(u) is the degree of vertex uu in GG and E(G)E(G) is the set of edges of GG. Despite many attempts in the last few years, it is still an open problem to characterize trees with minimal ABCABC index. In this paper, we present an efficient approach of computing trees with minimal ABC index, by considering the degree sequences of trees and some known properties of the graphs with minimal ABCABC index. The obtained results disprove some existing conjectures end suggest new ones to be set

    On reformulated zagreb indices with respect to tricyclic graphs

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    The authors Milic˘\breve{c}evicˊ\acute{c} et al. introduced the reformulated Zagreb indices, which is a generalization of classical Zagreb indices of chemical graph theory. In the paper, we characterize the extremal properties of the first reformulated Zagreb index. We first introduce some graph operations which increase or decrease this index. Furthermore, we will determine the extremal tricyclic graphs with minimum and maximum the first Zagreb index by these graph operations.Comment: 8 pages,2 figure

    Adapting the Interrelated Two-way Clustering method for Quantitative Structure-Activity Relationship (QSAR) Modeling of a Diverse Set of Chemical Compounds

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    Interrelated Two-way Clustering (ITC) is an unsupervised clustering method developed to divide samples into two groups in gene expression data obtained through microarrays, selecting important genes simultaneously in the process. This has been found to be a better approach than conventional clustering methods like K-means or self-organizing map for the scenarios when number of samples much smaller than number of variables (n<<p). In this paper we used the ITC approach for classification of a diverse set of 508 chemicals regarding mutagenicity. A large number of topological indices (TIs), 3-dimensional, and quantum chemical descriptors, as well as atom pairs (APs) have been used as explanatory variables. In this paper, ITC has been used only for predictor selection, after which ridge regression is employed to build the final predictive model. The proper leave-one-out (LOO) method of cross-validation in this scenario is to take as holdout each of the 508 compounds before predictor thinning and compare the predicted values with the experimental data. ITC based results obtained here are comparable to those developed earlier

    Computing the Mostar index in networks with applications to molecular graphs

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    Recently, a bond-additive topological descriptor, named as the Mostar index, has been introduced as a measure of peripherality in networks. For a connected graph GG, the Mostar index is defined as Mo(G)=βˆ‘e=uv∈E(G)∣nu(e)βˆ’nv(e)∣Mo(G) = \sum_{e=uv \in E(G)} |n_u(e) - n_v(e)|, where for an edge e=uve=uv we denote by nu(e)n_u(e) the number of vertices of GG that are closer to uu than to vv and by nv(e)n_v(e) the number of vertices of GG that are closer to vv than to uu. In this paper, we generalize the definition of the Mostar index to weighted graphs and prove that the Mostar index of a weighted graph can be computed in terms of Mostar indices of weighted quotient graphs. As a consequence, we show that the Mostar index of a benzenoid system can be computed in sub-linear time with respect to the number of vertices. Finally, our method is applied to some benzenoid systems and to a fullerene patch

    Topological indices of k-th subdivision and semi total point graphs

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    Graph theory has provided a very useful tool, called topological indices which are a number obtained from the graph GG with the property that every graph HH isomorphic to GG, value of a topological index must be same for both GG and HH. In this article, we present exact expressions for some topological indices of k-th subdivision graph and semi total point graphs respectively, which are a generalization of ordinary subdivision and semi total graph for kβ‰₯1k\ge 1.Comment: 14 page

    Computing weighted Szeged and PI indices from quotient graphs

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    The weighted Szeged index and the weighted vertex-PI index of a connected graph GG are defined as wSz(G)=βˆ‘e=uv∈E(G)(deg(u)+deg(v))nu(e)nv(e)wSz(G) = \sum_{e=uv \in E(G)} (deg (u) + deg (v))n_u(e)n_v(e) and wPIv(G)=βˆ‘e=uv∈E(G)(deg(u)+deg(v))(nu(e)+nv(e))wPI_v(G) = \sum_{e=uv \in E(G)} (deg(u) + deg(v))( n_u(e) + n_v(e)), respectively, where nu(e)n_u(e) denotes the number of vertices closer to uu than to vv and nv(e)n_v(e) denotes the number of vertices closer to vv than to uu. Moreover, the weighted edge-Szeged index and the weighted PI index are defined analogously. As the main result of this paper, we prove that if GG is a connected graph, then all these indices can be computed in terms of the corresponding indices of weighted quotient graphs with respect to a partition of the edge set that is coarser than the Ξ˜βˆ—\Theta^*-partition. If GG is a benzenoid system or a phenylene, then it is possible to choose a partition of the edge set in such a way that the quotient graphs are trees. As a consequence, it is shown that for a benzenoid system the mentioned indices can be computed in sub-linear time with respect to the number of vertices. Moreover, closed formulas for linear phenylenes are also deduced. However, our main theorem is proved in a more general form and therefore, we present how it can be used to compute some other topological indices

    Further results on degree based topological indices of certain chemical networks

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    There are various topological indices such as degree based topological indices, distance based topological indices and counting related topological indices etc. These topological indices correlate certain physicochemical properties such as boiling point, stability of chemical compounds. In this paper, we compute the sum-connectivity index and multiplicative Zagreb indices for certain networks of chemical importance like silicate networks, hexagonal networks, oxide networks, and honeycomb networks. Moreover, a comparative study using computer-based graphs has been made to clarify their nature for these families of networks.Comment: Submitte
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