The concept of minimal resolving partition and resolving set plays a pivotal role in diverse areas such as robot navigation, networking, optimization, mastermind games and coin weighing. It is hard to compute exact values of partition dimension for a graphic metric space, (G, dG) and networks. In this article, we give the sharp upper bounds and lower bounds for the partition dimension of generalized Möbius ladders, Mm, n, for all n≥3 and m≥2.

Growth series is an important invariant associated with group or monoid which classifies all the words of group or monoid. Therefore, the growth series of braid monoids and Hecke algebras in Artin’s generators is presented in many scholarly published articles. The growth series of braid monoids M B 3 and M B 4 in band generators is known. In this work, we compute the complete presentation of braid monoid M B 5 in band generators by solving all the ambiguities of M B 5 . The words on the left-hand of each relation are reducible words, and the words on the right-hand side are canonical words. We partially find the growth series Q ∗ 5 of reducible words. Then, we construct a linear system for canonical words of M B 5 in band presentation and compute the corresponding growth series. We also find the growth rate of growth series of M B 5 in band generators.

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 ${\Delta }^{2k+1}=x_1^{2k+2}{x}_2{x}_1^2{x}_2^2{x}_1^2\cdots x_2^2{x}_1^2{x}_1^2$ , ${\Delta }^{2k+1}{x}_2$ , and ${\Delta }^{2k+1}{x}_1$ , where $\Delta$ is the Garside element $x_1{x}_2{x}_1$ , and which are three out of all six classes of the general braid $x_1{x}_2{x}_1{x}_2\cdots$ with n factors.