<abstract><p>In this paper, we determine the exact metric and fault-tolerant metric dimension of the benzenoid tripod structure. We also computed the generalized version of this parameter and proved that all the parameters are constant. Resolving set $ {L} $ is an ordered subset of nodes of a graph $ {C} $, in which each vertex of $ {C} $ is distinctively determined by its distance vector to the nodes in $ {L} $. The cardinality of a minimum resolving set is called the metric dimension of $ {C} $. A resolving set $ L_{f} $ of $ {C} $ is fault-tolerant if $ {L}_{f}\setminus{b} $ is also a resolving set, for every $ {b} $ in $ {L}_{f}. $ Resolving set allows to obtain a unique representation for chemical structures. In particular, they were used in pharmaceutical research for discovering patterns common to a variety of drugs. The above definitions are based on the hypothesis of chemical graph theory and it is a customary depiction of chemical compounds in form of graph structures, where the node and edge represents the atom and bond types, respectively.</p></abstract>

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In this paper, we seek to present some new identities for the elementary symmetric polynomials and use these identities to construct new explicit formulas for the Legendre polynomials. First, we shed light on the variable nature of elementary symmetric polynomials in terms of repetition and additive inverse by listing the results related to these. Sequentially, we have proven an important formula for the Legendre polynomials, in which the exponent moves one walk instead of twice as known. The importance of this formula appears throughout presenting Vieta’s formula for the Legendre polynomials in terms of their zeros and the results mentioned therein. We provide new identities for the elementary symmetric polynomials of the zeros of the Legendre polynomials. Finally, we propose the relationship between elementary symmetric polynomials for the zeros of P n x and the zeros of P n − 1 x .

The main aim of this study is to define degenerate Genocchi polynomials and numbers of the second kind by using logarithmic functions, and to investigate some of their analytical properties and some applications. For this purpose, many formulas and relations for these polynomials, including some implicit summation formulas, differentiation rules and correlations with the earlier polynomials by utilizing some series manipulation methods, are derived. Additionally, as an application, the zero values of degenerate Genocchi polynomials and numbers of the second kind are presented in tables and multifarious graphical representations for these zero values are shown.

The main purpose of this paper is to consider q-sine-based and q-cosine-Based q-Fubini polynomials and is to investigate diverse properties of these polynomials. Furthermore, multifarious correlations including q-analogues of the Genocchi, Euler and Bernoulli polynomials, and the q-Stirling numbers of the second kind are derived. Moreover, some approximate zeros of the q-sine-based and q-cosine-Based q-Fubini polynomials in a complex plane are examined, and lastly, these zeros are shown using figures.