Extremal Triangular Chain Graphs for Bond Incident Degree (BID) Indices

A general expression for calculating the bond incident degree (BID) indices of certain triangular chain graphs is derived. The extremal triangular chain graphs with respect to several well known BID indices are also characterized over a particular collection of triangular chain graphs.Comment: 15 pages, 4 Figure

A new approximation to the geometric-arithmetic index

The concept of geometric-arithmetic index was introduced in the chemical graph theory recently, but it has shown to be useful. The aim of this paper is to obtain new inequalities involving the geometric-arithmetic index $GA_1$ and characterize graphs extremal with respect to them.Comment: 13 page

New lower bounds for the Geometric-Arithmetic index

The concept of geometric-arithmetic index was introduced in the chemical graph theory recently, but it has shown to be useful. The aim of this paper is to obtain new inequalities involving the geometric-arithmetic index $GA_1$ and characterize graphs extremal with respect to them. Our main results provide lower bounds $GA_1(G)$ involving just the minimum and the maximum degree of the graph $G$.Comment: 12 page

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

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

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

Inverse sum indeg energy of graphs

Suppose G is an n-vertex simple graph with vertex set {v1,..., vn} and d(i), i = 1,..., n, is the degree of vertex vi in G. The ISI matrix S(G) = [sij] of G is a square matrix of order n and is defined by sij = d(i)d(j)/d(i)+d(j) if the vertices vi and vj are adjacent and sij = 0 otherwise. The S-eigenvalues of G are the eigenvalues of its ISI matrix S(G). Recently the notion of inverse sum indeg (henceforth, ISI) energy of graphs is introduced and is defined as the sum of absolute values of S-eigenvalues of graph G. We give ISI energy formula of some graph classes. We also obtain some bounds for ISI energy of graphs

Geometric-Arithmetic index and line graph

The concept of geometric-arithmetic index was introduced in the chemical graph theory recently, but it has shown to be useful. The aim of this paper is to obtain new inequalities involving the geometric-arithmetic index $GA_1$ and characterize graphs extremal with respect to them. Besides, we prove inequalities involving the geometric-arithmetic index of line graphs