Topological indices are graph invariants computed by the distance or degree of vertices of the molecular graph. In chemical graph theory, topological indices have been successfully used in describing the structures and predicting certain physicochemical properties of chemical compounds. In this paper, we propose a definition of generalized bridge molecular graphs that can model more kinds of long chain polymerization products than the bridge molecular graphs, and provide some results of the edge versions of atom-bond connectivity ( $AB{C}_e$ ) and geometric arithmetic ( $G{A}_e$ ) indices for some generalized bridge molecular graphs, which have regular, periodic and symmetrical structures. The results of this paper offer promising prospects in the applications for chemical and material engineering, especially in chemical industry research.

Let $G = (V(G), E(G))$ be a connected graph and $\Pi = \{S_{1}, S_2, \dots, S_{k}\}$ be a $k$-partition of $V(G)$. The representation $r(v|\Pi)$ of a vertex $v$ with respect to $\Pi$ is the vector $(d(v, S_{1}), d(v, S_2), \dots, d(v, S_{k}))$, where $d(v, S_{i}) = \min\{d(v, s_{i})\mid s_{i}\in S_{i}\}$. The partition $\Pi$ is called a resolving partition of $G$ if $r(u|\Pi)\neq r(v|\Pi)$ for all distinct $u, v\in V(G)$. The partition dimension of $G$, denoted by $pd(G)$, is the cardinality of a minimum resolving partition of $G$. In this paper, we calculate the partition dimension of two $(4, 6)$-fullerene graphs. We also give conjectures on the partition dimension of two $(3, 6)$-fullerene graphs.

The analysis of networks and graphs through topological descriptors carries out a useful role to derive their underlying topologies. This process has been widely used in biomedicine, cheminformatics, and bioinformatics, where assessments based on graph invariants have been made available for effectively communicating with the various challenging schemes. In the studies of quantitative structure-activity relationships (QSARs) and quantitative structure-property relationships (QSPRs), graph invariants are used to approximate the biological activities and properties of chemical compounds. In this paper, we give the results related to the eccentric-connectivity index, connective eccentricity index, total-eccentricity index, average eccentricity index, Zagreb eccentricity indices, eccentric geometric-arithmetic index, eccentric atom-bond connectivity index, eccentric adjacency index, modified eccentric-connectivity index, eccentric distance sum, Wiener index, Harary index, hyper-Wiener index and degree distance index of a new graph operation named as “subdivision vertex-edge join” of three graphs.

The general sum-connectivity index is a molecular descriptor defined as $\chi_{\alpha}(X)=\sum_{xy\in E(X)}(d_{X}(x)+d_{X}(y))^{\alpha}$ , where $d_{X}(x)$ denotes the degree of a vertex $x\in X$ , and α is a real number. Let X be a graph; then let $R(X)$ be the graph obtained from X by adding a new vertex $x_{e}$ corresponding to each edge of X and joining $x_{e}$ to the end vertices of the corresponding edge $e\in E(X)$ . In this paper we obtain the lower and upper bounds for the general sum-connectivity index of four types of graph operations involving R-graph. Additionally, we determine the bounds for the general sum-connectivity index of line graph $L(X)$ and rooted product of graphs.

Let G =(VG,EG) be a simple and connected graph. The eccentric connectivity index of G is represented as ξc(G)=∑xϵVGdegG(x)ecG(x), where degG(x) and ecG(x) represent the degree and the eccentricity of x, respectively. The eccentric adjacency index of G is represented as ξad(G)=∑xϵVGSG(x)/ecG(x), where SG(x) is the sum of degrees of neighbors of x. In this article, we determine the trees with the smallest eccentric connectivity index when bipartition size, independence number and domination number are given. Furthermore, we discuss the trees with the largest eccentric adjacency index when bipartition size, matching number and independence number are given.