Abstract. 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.

The association of M-polynomial to chemical compounds and chemical networks is a relatively new idea, and it gives good results about the topological indices. These results are then used to correlate the chemical compounds and chemical networks with their chemical properties and bioactivities. In this paper, an effort is made to compute the general form of the M-polynomials for two classes of dendrimer nanostars and four types of nanotubes. These nanotubes have very nice symmetries in their structural representations, which have been used to determine the corresponding M-polynomials. Furthermore, by using the general form of M-polynomial of these nanostructures, some degree-based topological indices have been computed. In the end, the graphical representation of the M-polynomials is shown, and a detailed comparison between the obtained topological indices for aforementioned chemical structures is discussed.

Neural networks in which communication works only among the neighboring units are called cellular neural networks (CNNs). These are used in analyzing 3D surfaces, image processing, modeling biological vision, and reducing nonvisual problems of geometric maps and sensory-motor organs. Topological indices (TIs) are mathematical models of the (molecular) networks or structures which are presented in the form of numerical values, constitutional formulas, or numerical functions. These models predict the various chemical or structural properties of the under-study networks. We now consider analogous graph invariants, based on the second connection number of vertices, called Zagreb connection indices. The main objective of this paper is to compute these connection indices for the cellular neural networks (CNNs). In order to find their efficiency, a comparison among the obtained indices of CNN is also performed in the form of numerical tables and 3D plots.

In this paper, we introduce the concept of spanning simplicial complexes ∆ s (G) associated to a simple finite connected graph G. We give the characterization of all spanning trees of the uni-cyclic graph U n,m . In particular, we give the formula for computing the Hilbert series and h-vector of the Stanley Riesner ring k ∆ s (U n,m ) . Finally, we prove that the spanning simplicial complex ∆ s (U n,m ) is shifted hence ∆ s (U n,m ) is shellable.

Nekrasov-Okounkov identity gives a product representation of the sum over partitions of a certain function of partition hook length. In this paper we give several generalizations of the Nekrasov-Okounkov identity using the cyclic symmetry of the topological vertex.

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The general sum-connectivity index is a molecular descriptor defined as , where denotes the degree of a vertex , and α is a real number. Let X be a graph; then let be the graph obtained from X by adding a new vertex corresponding to each edge of X and joining to the end vertices of the corresponding edge . 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 and rooted product of graphs.

Phenylenes are a class of conjugated hydrocarbons composed of a special arrangement of six- and four-membered rings. In this paper, the expected values of the atom-bond connectivity (ABC), geometric-arithmetic (GA), and Zagreb indices of this class of conjugated hydrocarbons have been determined. At the end, the comparisons with respect to the random phenylene chains among the expected values of these indices, have been determined explicitly. The graphical profiles of these indices have been shown in order to support our results.