Hex-derived network has a variety of useful applications in pharmacy, electronics, and networking. In this paper, we give general form of the M-polynomial of the hex-derived networksHDN1[n] and HDN2[n], which came out of n-dimensional hexagonal mesh. We also give closed forms of several degree-based topological indices associated to these networks.

In this paper we introduce the walk polynomial to find the number of walks of different lengths in a simple connected graph. We also give the walk polynomial of the bipartite, star, wheel, and gear graphs in closed forms.

Degree and distance-based graph polynomials are important not only as graph invariants but also for their applications in physics, chemistry, and pharmacy. The present paper is concerned with the Hosoya and Schultz polynomials of n-bilinear straight pentachain. It was observed that computing these polynomials directly by definitions is extremely difficult. Thus, we follow the divide and conquer rule and introduce the idea of base polynomials. We actually split the vertices into disjoint classes and then wrote the number of paths in terms of polynomials for each class, which ultimately serve as bases for Hosoya and Schultz polynomials. We also recover some indices from these polynomials. Finally, we give an example to show how these polynomials actually work.

This paper is concerned with the combinatorial facts of the lattice graphs of ${\mathbb{Z}}_{p_1\times p_2\times \cdots \times p_m}$ , ${\mathbb{Z}}_{p_1^{m_1}\times p_2^{m_2}}$ , and ${\mathbb{Z}}_{p_1^{m_1}\times p_2^{m_2}\times p_3^1}$ . We show that the lattice graph of ${\mathbb{Z}}_{p_1\times p_2\times \cdots \times p_m}$ is realizable as a convex polytope. We also show that the diameter of the lattice graph of ${\mathbb{Z}}_{p_1^{m_1}\times p_2^{m_2}\times \cdots \times p_r^{m_r}}$ is $\sum _{i=1}^r{m}_i$ and its girth is 4.

We introduce a polygonal cylinder \( C_{m,n} \), using the Cartesian product of paths \( P_m \) and \( P_n \) and using topological identification of vertices and edges of two opposite sides of \( P_m \times P_n \), and give its Hosoya polynomial, which, depending on odd and even \( m \), is covered in seven separate cases.