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

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

On the maximum and minimum multiplicative Zagreb indices of graphs with given number of cut edges

For a molecular graph, the first multiplicative Zagreb index $\Pi_1$ is equal to the product of the square of the degree of the vertices, while the second multiplicative Zagreb index $\Pi_2$ is equal to the product of the endvertex degree of each edge over all edges. Denote by $\mathbb{G}_{n,k}$ the set of graphs with $n$ vertices and $k$ cut edges. In this paper, we explore graphs in terms of a number of cut edges. In addition, the maximum and minimum multiplicative Zagreb indices of graphs with given number of cut edges are provided. Furthermore, we characterize graphs with the largest and smallest $\Pi_1(G)$ and $\Pi_2(G)$ in $\mathbb{G}_{n,k}$, and our results extend and enrich some known conclusions

On the sharp upper and lower bounds of multiplicative Zagreb indices of graphs with connectivity at most k

For a (molecular) graph, the first multiplicative Zagreb index $\prod_1(G)$ is the product of the square of every vertex degree, and the second multiplicative Zagreb index $\prod_2(G)$ is the product of the products of degrees of pairs of adjacent vertices. In this paper, we explore graphs in terms of (edge) connectivity. The maximum and minimum values of $\prod_1(G)$ and $\prod_2(G)$ of graphs with connectivity at most $k$ are provided. In addition, the corresponding extremal graphs are characterized, and our results extend and enrich some known conclusions

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

The first Zagreb index of a graph $G$ 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 $\mathbb{V}_{n,k}$, where $\mathbb{V}_{n, k}$ denotes the set of all $n$-vertex graphs with $k$ 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

Bond Incident Degree (BID) Indices of Polyomino Chains: A Unified Approach

This work is devoted to establish a general expression for calculating the bond incident degree (BID) indices of polyomino chains and to characterize the extremal polyomino chains with respect to several well known BID indices. From the derived results, all the results of [M. An, L. Xiong, Extremal polyomino chains with respect to general Randi\'{c} index, \textit{J. Comb. Optim.} (2014) DOI 10.1007/s10878-014-9781-6], [H. Deng, S. Balachandran, S. K. Ayyaswamy, Y. B. Venkatakrishnan, The harmonic indices of polyomino chains, \textit{Natl. Acad. Sci. Lett.} \textbf{37}(5), (2014) 451-455], [Z. Yarahmadi, A. R. Ashrafi and S. Moradi, Extremal polyomino chains with respect to Zagreb indices, \textit{Appl. Math. Lett.} \textbf{25} (2012) 166-171], and also some results of [J. Rada, The linear chain as an extremal value of VDB topological indices of polyomino chains, \textit{Appl. Math. Sci.} \textbf{8}, (2014) 5133-5143], [A. Ali, A. A. Bhatti, Z. Raza, Some vertex-degree-based topological indices of polyomino chains, \textit{J. Comput. Theor. Nanosci.} \textbf{12}(9), (2015) 2101-2107] are obtained as corollaries.Comment: 17 pages, 3 figure

Further results on degree based topological indices of certain chemical networks

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

Minimizing Degree-based Topological Indices for Trees with Given Number of Pendent Vertices + Erratum

We derive sharp lower bounds for the first and the second Zagreb indices ($M_1$ and $M_2$ respectively) for trees and chemical trees with the given number of pendent vertices and find optimal trees. $M_1$ is minimized by a tree with all internal vertices having degree 4, while $M_2$ is minimized by a tree where each "stem" vertex is incident to 3 or 4 pendent vertices and one internal vertex, while the rest internal vertices are incident to 3 other internal vertices. The technique is shown to generalize to the weighted first Zagreb index, the zeroth order general Randi\'{c} index, as long as to many other degree-based indices. Later the erratum was added: Theorem 3 says that the second Zagreb index $M_2$ cannot be less than $11n-27$ for a tree with $n\ge 8$ pendent vertices. Yet the tree exists with $n=8$ vertices (the two-sided broom) violating this inequality. The reason is that the proof of Theorem 3 relays on a tacit assumption that an index-minimizing tree contains no vertices of degree 2. This assumption appears to be invalid in general. In this erratum we show that the inequality $M_2 \ge 11n-27$ still holds for trees with $n\ge 9$ vertices and provide the valid proof of the (corrected) Theorem 3.Comment: Original paper: 15 pages, 2 figures. Erratum: 8 pages, 2 figures. Professor Tam'as R'eti contributed to the erratu

New Lower Bounds for the First Variable Zagreb Index

The aim of this paper is to obtain new sharp inequalities for a large family of topological indices, including the first variable Zagreb index $M_1^\alpha$, and to characterize the set of extremal graphs with respect to them. Our main results provide lower bounds on this family of topological indices involving just the minimum and the maximum degree of the graph. These inequalities are new even for the first Zagreb, the inverse and the forgotten indices.Comment: 17 page

Multiplicative Zagreb indices of cacti

Let $\prod(G)$ be Multiplicative Zagreb index of a graph G. A connected graph is a cactus graph if and only if any two of its cycles have at most one vertex in common, which has been the interest of researchers in the filed of material chemistry and graph theory. In this paper, we use a new tool to the obtain upper and lower bounds of $\prod(G)$ for all cactus graphs and characterize the corresponding extremal graphs.Comment: Discrete Mathematics, Algorithms and Applications(accpeted) 201

On extremal results of multiplicative Zagreb indices of trees with given distance kk-domination number

The first multiplicative Zagreb index $\Pi_1$ of a graph $G$ is the product of the square of every vertex degree, while the second multiplicative Zagreb index $\Pi_2$ is the product of the products of degrees of pairs of adjacent vertices. In this paper, we give sharp lower bound for $\Pi_1$ and upper bound for $\Pi_2$ of trees with given distance $k$-domination number, and characterize those trees attaining the bounds