Browsing by Author "Sardar, Muhammad Shoaib"

Now showing 1 - 4 of 4

Base Polynomials for Degree and Distance Based Topological Invariants of N-Bilinear Straight Pentachain
(Taylor & Francis Ltd, 2021) Nizami, Abdul Rauf; Shabbir, Khurram; Sardar, Muhammad Shoaib; Qasim, Muhammad; Cancan, Murat; Ediz, Suleyman
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.
On Modified Eccentric Connectivity Index of Namn Nanotube
(Analytic Publ Co, 2020) Sajjad, Wasim; Sardar, Muhammad Shoaib; Cancan, Murat; Ediz, Suleyman; Baig, Abdul Qudair
Eccentricity based topological indices have attracted large attraction in the field of chemical graph theory. Eccentricity based topological indices are prominent due to their wide range applications. With the help of eccentricity based topological indices we can predict excellent accuracy rate in certain biological activities of diverse nature as com-pared to other indices. Carbon nanotubes are the cylindrical molecules that consist of rolled up sheets of carbon atoms. In this paper we consider the chemical graph of NA(m)(n) nanotube and compute an important eccentricity based topological index called Modified eccentric connectivity index.
Resistance Distance in Some Classes of Rooted Product Graphs Obtained by Laplacian Generalized Inverse Method
(Taylor & Francis Ltd, 2021) Sardar, Muhammad Shoaib; Alaeiyan, Mehdi; Farahani, Mohammad Reza; Cancan, Murat; Ediz, Suleyman
In mathematics, a graph product is a binary operation on a graph. Graph products have been extensively researched and have many important applications in many fields. Here we discuss one graph-theoretical product. Let H be a labeled graph on n vertices and let G be a rooted graph. Denote by H G the graph obtained by identifying the root vertex of the ith copy of G with the ith vertex of H. H G is called by the rooted product of H by G [C. Godsil, B. D. McKay, A new graph product and its spectrum, Bull. Aust. Math. Soc. 18 (1978)]. The resistance distance between two vertices i and j of a graph G is defined as the effective resistance between the two vertices when a unit resistor replaces each edge of G. Let H-k;n, C-m, S-k, P-k and K-u be the Harary, cycle, star, path and complete graphs respectively. In this paper, the symmetric {1}-inverses of Laplacian matrices for graphs (H-k;n circle C-m), (H-k;n circle K-u), (C-n circle S-k) and (C-n circle P-k) are studied, based on which the resistance distances of any two vertices in these graphs can be obtained. In addition, some examples are provided as applications that illustrate the functionality of the suggested method.
The Study of the B-Choromatic Number of Some Classes of Fractal Graphs
(Taru Publications, 2022) Sattar, Tayyiba; Sardar, Muhammad Shoaib; Alaeiyan, Mehdi; Farahani, Moahmmad Reza; Cancan, Murat; Tas, Ziyattin
In graph coloring, labels are assigned to graph elements according to certain constraints. Colors are a special case of graph labeling as well as in practical applications, graph coloring also poses some theoretical challenges. A topic related to graph coloring will be discussed in this study, i.e., b-chromatic number. In proper coloring, edges, vertices, or both of them are colored so that they are distinct from one another. A b-coloring of m colors of a graph G is similar to proper coloring in which at least one vertex from each color class is connected to (m-1) other colors. The b-chromatic number of a graph G is the greatest positive number k such that G admits a b-coloring with k colors and is represented by phi(G). Fractals are geometric objects that are self-similar at multiple scales and their geometric measurements are different from fractal measurements. In this paper, we will evaluate the b-chromatic number of Fractal type graphs, i.e., Sierpinski network S(n; Kk) (where Kk is a complete graph of order k) and Sierpinski gasket network S(n). Firstly, we will compute the b-chromatic number of S(n; K3), S(n; K4) and S(n; K5) for n >= 2. After that, we will generalize the result for the Sierpinski network of complete graph Kk. In addition, we will also determine the b-choromatic number of Sierpinski gasket graph S(n). As an application, we will also determine the b-chromatic number of Sierpinski graph of house graph.