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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.
Abdul Rauf Nizami; Khurram Shabbir; Muhammad Shoaib Sardar; Muhammad Qasim; Murat Cancan; Süleyman Ediz. Base polynomials for degree and distance based topological invariants of n-bilinear straight pentachain. Journal of Information and Optimization Sciences 2021, 1 -17.
AMA StyleAbdul Rauf Nizami, Khurram Shabbir, Muhammad Shoaib Sardar, Muhammad Qasim, Murat Cancan, Süleyman Ediz. Base polynomials for degree and distance based topological invariants of n-bilinear straight pentachain. Journal of Information and Optimization Sciences. 2021; ():1-17.
Chicago/Turabian StyleAbdul Rauf Nizami; Khurram Shabbir; Muhammad Shoaib Sardar; Muhammad Qasim; Murat Cancan; Süleyman Ediz. 2021. "Base polynomials for degree and distance based topological invariants of n-bilinear straight pentachain." Journal of Information and Optimization Sciences , no. : 1-17.
This paper is concerned with the combinatorial facts of the lattice graphs of Z p 1 × p 2 × ⋯ × p m , Z p 1 m 1 × p 2 m 2 , and Z p 1 m 1 × p 2 m 2 × p 3 1 . We show that the lattice graph of Z p 1 × p 2 × ⋯ × p m is realizable as a convex polytope. We also show that the diameter of the lattice graph of Z p 1 m 1 × p 2 m 2 × ⋯ × p r m r is ∑ i = 1 r m i and its girth is 4.
Jia-Bao Liu; Mobeen Munir; Qurat-Ul-Ain Munir; Abdul Rauf Nizami. Some Metrical Properties of Lattice Graphs of Finite Groups. Mathematics 2019, 7, 398 .
AMA StyleJia-Bao Liu, Mobeen Munir, Qurat-Ul-Ain Munir, Abdul Rauf Nizami. Some Metrical Properties of Lattice Graphs of Finite Groups. Mathematics. 2019; 7 (5):398.
Chicago/Turabian StyleJia-Bao Liu; Mobeen Munir; Qurat-Ul-Ain Munir; Abdul Rauf Nizami. 2019. "Some Metrical Properties of Lattice Graphs of Finite Groups." Mathematics 7, no. 5: 398.
Khovanov homology is a categorication of the Jones polynomial. It consists of graded chain complexes which, up to chain homotopy, are link invariants, and whose graded Euler characteristic is equal to the Jones polynomial of the link. In this article we give some Khovanov homology groups of 3-strand braid links Δ 2 k + 1 = x 1 2 k + 2 x 2 x 1 2 x 2 2 x 1 2 ⋯ x 2 2 x 1 2 x 1 2 , Δ 2 k + 1 x 2 , and Δ 2 k + 1 x 1 , where Δ is the Garside element x 1 x 2 x 1 , and which are three out of all six classes of the general braid x 1 x 2 x 1 x 2 ⋯ with n factors.
Young Chel Kwun; Abdul Rauf Nizami; Mobeen Munir; Zaffar Iqbal; Dishya Arshad; Shin Min Kang. Khovanov Homology of Three-Strand Braid Links. Symmetry 2018, 10, 720 .
AMA StyleYoung Chel Kwun, Abdul Rauf Nizami, Mobeen Munir, Zaffar Iqbal, Dishya Arshad, Shin Min Kang. Khovanov Homology of Three-Strand Braid Links. Symmetry. 2018; 10 (12):720.
Chicago/Turabian StyleYoung Chel Kwun; Abdul Rauf Nizami; Mobeen Munir; Zaffar Iqbal; Dishya Arshad; Shin Min Kang. 2018. "Khovanov Homology of Three-Strand Braid Links." Symmetry 10, no. 12: 720.
Recently Berceanu and Iqbal proved that the growth rate of all the spherical Artin monoids is bounded above by 4. In this paper we compute the Hilbert series of the right-angled spherical Artin monoid $\begin{array}{} M({D}^{\infty}_{n}) \end{array} $ and graphically prove that growth rate is bounded by 4. We also discuss its recurrence relations and other main properties.
Zaffar Iqbal; Abdul Rauf Nizami; Mobeen Munir; Amlish Rabia; Shin Min Kang. On right-angled spherical Artin monoid of type Dn. Open Physics 2018, 16, 441 -447.
AMA StyleZaffar Iqbal, Abdul Rauf Nizami, Mobeen Munir, Amlish Rabia, Shin Min Kang. On right-angled spherical Artin monoid of type Dn. Open Physics. 2018; 16 (1):441-447.
Chicago/Turabian StyleZaffar Iqbal; Abdul Rauf Nizami; Mobeen Munir; Amlish Rabia; Shin Min Kang. 2018. "On right-angled spherical Artin monoid of type Dn." Open Physics 16, no. 1: 441-447.
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.
Shin Min Kang; Waqas Nazeer; Manzoor Ahmad Zahid; Abdul Rauf Nizami; Adnan Aslam; Mobeen Munir. M-polynomials and topological indices of hex-derived networks. Open Physics 2018, 16, 394 -403.
AMA StyleShin Min Kang, Waqas Nazeer, Manzoor Ahmad Zahid, Abdul Rauf Nizami, Adnan Aslam, Mobeen Munir. M-polynomials and topological indices of hex-derived networks. Open Physics. 2018; 16 (1):394-403.
Chicago/Turabian StyleShin Min Kang; Waqas Nazeer; Manzoor Ahmad Zahid; Abdul Rauf Nizami; Adnan Aslam; Mobeen Munir. 2018. "M-polynomials and topological indices of hex-derived networks." Open Physics 16, no. 1: 394-403.
Muhammad Idrees; HongBin Ma; Numan Amin; Abdul Rauf Nizami; Zaffar Iqbal; Saiid Ali. Several Topological Invariants of Generalized Möbius Ladder. 2018 37th Chinese Control Conference (CCC) 2018, 1 .
AMA StyleMuhammad Idrees, HongBin Ma, Numan Amin, Abdul Rauf Nizami, Zaffar Iqbal, Saiid Ali. Several Topological Invariants of Generalized Möbius Ladder. 2018 37th Chinese Control Conference (CCC). 2018; ():1.
Chicago/Turabian StyleMuhammad Idrees; HongBin Ma; Numan Amin; Abdul Rauf Nizami; Zaffar Iqbal; Saiid Ali. 2018. "Several Topological Invariants of Generalized Möbius Ladder." 2018 37th Chinese Control Conference (CCC) , no. : 1.
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.
Abdul Rauf Nizami; Afshan Perveen; Waqas Nazeer; Mahnoor Baqir. WALK POLYNOMIAL: A New Graph Invariant. Applied Mathematics and Nonlinear Sciences 2018, 3, 321 -330.
AMA StyleAbdul Rauf Nizami, Afshan Perveen, Waqas Nazeer, Mahnoor Baqir. WALK POLYNOMIAL: A New Graph Invariant. Applied Mathematics and Nonlinear Sciences. 2018; 3 (1):321-330.
Chicago/Turabian StyleAbdul Rauf Nizami; Afshan Perveen; Waqas Nazeer; Mahnoor Baqir. 2018. "WALK POLYNOMIAL: A New Graph Invariant." Applied Mathematics and Nonlinear Sciences 3, no. 1: 321-330.