Many properties of special polynomials, such as recurrence relations, sum formulas, and symmetric properties have been studied in the literature with the help of generating functions and their functional equations. In this paper, using the $(p,q)$–Fibonacci polynomials, $(p,q)$–Lucas polynomials, and Changhee numbers, we define the $(p,q)$–Fibonacci–Changhee polynomials and $(p,q)$–Lucas–Changhee polynomials, respectively. We obtain some important identities and relations of these newly established polynomials by using their generating functions and functional equations. Then, we generalize the $(p,q)$–Fibonacci–Changhee polynomials and the $(p,q)$–Lucas–Changhee polynomials called generalized $(p,q)$–Fibonacci–Lucas–Changhee polynomials. We derive a determinantal representation for the generalized $(p,q)$–Fibonacci–Lucas–Changhee polynomials in terms of the special Hessenberg determinant. Finally, we give a new recurrent relation of the $(p,q)$–Fibonacci–Lucas–Changhee polynomials.

The distance between two vertices of a simple connected graph G, denoted as $d(u,v)$, is the length of the shortest path from u to v and is always symmetrical. An ordered subset $W=\left\{w_1,w_2,w_3,\cdots ,w_k\right\}$ of $V\left(G\right)$ is a resolving set for G, if for ∀$u,v\in V\left(G\right)$, there exists $w_i\in W$ ∋ $d(u,w_i)\ne d(v,w_i)$. A resolving set with minimal cardinality is called the metric basis. The metric dimension of G is the cardinality of metric basis of G and is denoted as $dim\left(G\right)$. For the graph $G_1=(V_1,E_{1,})$ and $G_2=(V_2,E_2)$, their join is denoted by $G_1+G_2$. The vertex set of $G_1+G_2$ is $V_1\cup V_2$ and the edge set is $E=E_1\cup E_2\cup \left\{uv,u\in V_1,v\in V_2\right\}$. In this article, we show that the metric dimension of the join of two path graphs is unbounded because of its dependence on the size of the paths. We also provide a general formula to determine this metric dimension. We also develop algorithms to obtain metric dimensions and a metric basis for the join of path graphs, with respect to its symmetries.