Computation of vertex and edge resolvability of benzenoid tripod structure

Definition 1.1 Nadeem et al. (2020)

“Assume $C$ be an associated graph of chemical structure/network, whose vertex/node set we will denote with symbol $N(C)$ or simply $N$ , while $B(C)$ or $B$ is the edge/bond set, the shortest distance between two bonds $b_1,b_2\in N(C)$ denoted by $S_{b_1,b_2}$ , and calculated by counting the number of bonds while moving through the $b_1-b_2$ path. The distance between an edge $e=b_1{b}_2\in B(C)$ , and a node $b\in N(C)$ is counted by the relation $S_{e,b}=\mathit{min}\{S_{b_1,b},S_{b_2,b}\}$ .”

Definition 1.2 Deng et al. (mar 2021)

“A vertex $b$ of a connected associated graph $C$ of a chemical structure, differentiates nodes ( $b_1$ ) and edges ( $e_1$ ), if $S_{b_1,b}\ne S_{b,e_1}$ . A subset $F_m$ is a mixed resolving set if any different pair of components of $C$ are separated by a node of $F_m$ . The minimum number of nodes in mixed resolving set for $C$ is named the mixed metric dimension and is denoted by ${\mathit{dim}}_m(C)$ . It is also known as blended version of both metric (Slater, 1975) and edge metric dimension (Chartrand et al., 2000).”

Theorem 1.3 Kelenc et al. (Dec 2017)

“If ${\mathit{dim}}_m(C)$ is the mixed metric dimension, then ${\mathit{dim}}_m(C)=2$ , iff $C$ is a path $P_n$ .”

If $T\left({\theta }_1,{\theta }_2,{\theta }_3\right)$ is a benzenoid tripod graph with any of ${\theta }_1,{\theta }_2,{\theta }_3=2$ , then the mixed metric resolving set has minimum three members.

The order or the total amount of vertices in the benzenoid tripod’s associated graph with any of ${\theta }_1,{\theta }_2,{\theta }_3=2$ , are $4\left({\theta }_1+{\theta }_2+{\theta }_3\right)-8$ , and to look over the likelihood of mixed metric resolving set having three members in it, are $C\left(4\left({\theta }_1+{\theta }_2+{\theta }_3\right)-8,3\right)=\frac{\left(4\left({\theta }_1+{\theta }_2+{\theta }_3\right)-8\right)!}{2\times \left(4\left({\theta }_1+{\theta }_2+{\theta }_3\right)-11\right)!}$ . Over here we are focusing on three members in the set, from the result of Theorem 1.3 in the literature, the only two member of mixed metric resolving set is just for the basic graph f path. Now because of its computational cost of selecting mixed metric resolving set, for a graph, we can’t discover the precise number of mixed resolving sets, but from $\frac{\left(4\left({\theta }_1+{\theta }_2+{\theta }_3\right)-8\right)!}{2\times \left(4\left({\theta }_1+{\theta }_2+{\theta }_3\right)-11\right)!}$ -possibilities we selected a subset $F_m$ and described as: $F_m=\{a_1,a_{2{\theta }_3},b_{2{\theta }_1}\}$ . Proving of this claim that $F_m$ is also a subset having ability to be the candidate of mixed metric resolvability set of benzenoid tripod graph or $T\left({\theta }_1,{\theta }_2,{\theta }_3\right)$ , we will stick to 1.2’s definition. We will also look over the distinct placements or locations of each node and edge to meet the definition’s requirements, and the approach is explained above in Definition 1.2.

Positions $p\left(a_{\delta }|F_m\right)$ with respect to $F_m$ , for the nodes $a_{\delta }$ with $\delta =1,2,\dots ,2{\theta }_3$ , are described as: $$p\left(a_{\delta }|F_m\right)=\left(\delta -1,2{\theta }_3-\delta ,2\left({\theta }_3+{\theta }_1\right)-\delta \right)\text{.}$$

Positions $p\left(b_{\delta }|F_m\right)$ with respect to $F_m$ , for the nodes $b_{\delta }$ with $\delta =1,2,\dots ,2{\theta }_1$ , are described as: $$p\left(b_{\delta }|F_m\right)=\left(2{\theta }_3+\delta -1,\delta ,2{\theta }_1-\delta \right)\text{.}$$

Positions $p\left(c_{\delta }|F_m\right)$ with respect to $F_m$ , for the nodes $c_{\delta }$ with $\delta =1,2,\dots ,2{\theta }_2-1$ , are described as: $$p\left(c_{\delta }|F_m\right)=\left\{\begin{array}{cc}\left(2{\theta }_3+\delta -1,\delta +2,2{\theta }_1-1\right),\hfill & \mathrm{if}\delta =1\text{;}\hfill \\ \left(2{\theta }_3+\delta -1,\delta +2,2\left({\theta }_1-2\right)+\delta \right),\hfill & \mathrm{if}\delta =2\text{;}\hfill \\ \left(2{\theta }_3+\delta -3,\delta +2,2\left({\theta }_1-2\right)+\delta \right),\hfill & \mathrm{if}\delta =3,4,\dots ,2{\theta }_2-1\text{.}\hfill \end{array}\right.$$

Positions $p\left(a_{\delta }^{\prime }|F_m\right)$ with respect to $F_m$ , for the nodes $a_{\delta }^{\prime }$ with $\delta =1,2,\dots ,2{\theta }_3-3$ , are described as: $$p\left(a_{\delta }^{\prime }|F_m\right)=\left(\delta ,2{\theta }_3+1-\delta ,2\left({\theta }_3+{\theta }_1\right)-1-\delta \right)\text{.}$$

Positions $p\left(b_{\delta }^{\prime }|F_m\right)$ with respect to $F_m$ , for the nodes $b_{\delta }^{\prime }$ with $\delta =1,2,\dots ,2{\theta }_1-3$ , are described as: $$p\left(b_{\delta }^{\prime }|F_m\right)=\left(2{\theta }_3+\delta +1,\delta +4,2\left({\theta }_1-1\right)-\delta \right)\text{.}$$

Positions $p\left(c_{\delta }^{\prime }|F_m\right)$ with respect to $F_m$ , for the nodes $c_{\delta }^{\prime }$ with $\delta =1,2,\dots ,2{\theta }_2-1$ , are described as: $$p\left(c_{\delta }^{\prime }|F_m\right)=\left\{\begin{array}{cc}\left(2{\theta }_3-1,\delta +1,2{\theta }_1-1+\delta \right),\hfill & \mathrm{if}\delta =1\text{;}\hfill \\ \left(2\left({\theta }_3-2\right)+\delta ,\delta +1,2{\theta }_1-1+\delta \right),\hfill & \mathrm{if}\delta =2\text{;}\hfill \\ \left(2\left({\theta }_3-2\right)+\delta ,\delta +1,2{\theta }_1-3+\delta \right),\hfill & \mathrm{if}\delta =3,4,\dots ,2{\theta }_2-1\text{.}\hfill \end{array}\right.$$

Given above are the positions of nodes only, with respect to the chosen vertices in $F_m$ , for the fulfillment of definition of mixed resolving set we still need to investigates the edges or bonds of graph. Following are positions of bonds that will complete the definition.

Positions $p\left(a_{\delta }{a}_{\delta +1}|F_m\right)$ with regard to $F_m$ , for the bonds $a_{\delta }{a}_{\delta +1}$ with $\delta =1,2,\dots ,2{\theta }_3-1$ , are described as: $$p\left(a_{\delta }{a}_{\delta +1}|F_m\right)=\left\{\begin{array}{cc}\left(\delta -1,2{\theta }_3-1-\delta ,2\left({\theta }_3+{\theta }_1\right)-1-\delta \right),\hfill & \mathrm{if}\delta =1,2,\dots ,2{\theta }_3-2\text{;}\hfill \\ \left(\delta -1,2{\theta }_3-1-\delta ,2{\theta }_3\right),\hfill & \mathrm{if}\delta =2{\theta }_3-1\text{.}\hfill \end{array}\right.$$

Positions $p\left(b_{\delta }{b}_{\delta +1}|F_m\right)$ with regard to $F_m$ , for the bonds $b_{\delta }{b}_{\delta +1}$ with $\delta =1,2,\dots ,2{\theta }_1-1$ , are described as: $$p\left(b_{\delta }{b}_{\delta +1}|F_m\right)=\left(2{\theta }_3+\delta -1,\delta ,2{\theta }_1-1-\delta \right)\text{.}$$

Positions $p\left(c_{\delta }{c}_{\delta +1}|F_m\right)$ with regard to $F_m$ , for the bonds $c_{\delta }{c}_{\delta +1}$ with $\delta =1,2,\dots ,2{\theta }_2-2$ , are described as: $$p\left(c_{\delta }{c}_{\delta +1}|F_m\right)=\left\{\begin{array}{cc}\left(2{\theta }_3,\delta +2,2{\theta }_1-2\right),\hfill & \mathrm{if}\delta =1\text{;}\hfill \\ \left(2{\theta }_3,\delta +2,2\left({\theta }_1-2\right)+\delta \right),\hfill & \mathrm{if}\delta =2\text{;}\hfill \\ \left(2{\theta }_3+\delta -3,\delta +2,2\left({\theta }_1-2\right)+\delta \right),\hfill & \mathrm{if}\delta =3,4,\dots ,2{\theta }_2-1\text{.}\hfill \end{array}\right.$$

Positions $p\left(a_{\delta }^{\prime }{a}_{\delta +1}^{\prime }|F_m\right)$ with regard to $F_m$ , for the bonds $a_{\delta }^{\prime }{a}_{\delta +1}^{\prime }$ with $\delta =1,2,\dots ,2{\theta }_3-4$ , are described as: $$p\left(a_{\delta }^{\prime }{a}_{\delta +1}^{\prime }|F_m\right)=\left(\delta ,2{\theta }_3-\delta ,2\left({\theta }_3+{\theta }_1-1\right)-\delta \right)\text{.}$$

Positions $p\left(b_{\delta }^{\prime }{b}_{\delta +1}^{\prime }|F_m\right)$ with regard to $F_m$ , for the bonds $b_{\delta }^{\prime }{b}_{\delta +1}^{\prime }$ with $\delta =1,2,\dots ,2{\theta }_1-4$ , are described as: $$p\left(b_{\delta }^{\prime }{b}_{\delta +1}^{\prime }|F_m\right)=\left(2{\theta }_3+\delta +1,\delta +4,2{\theta }_1-3-\delta \right)\text{.}$$

Positions $p\left(c_{\delta }^{\prime }{c}_{\delta +1}^{\prime }|F_m\right)$ with regard to $F_m$ , for the bonds $c_{\delta }^{\prime }{c}_{\delta +1}^{\prime }$ with $\delta =1,2,\dots ,2{\theta }_2-2$ , are described as: $$p\left(c_{\delta }^{\prime }{c}_{\delta +1}^{\prime }|F_m\right)=\left\{\begin{array}{cc}\left(2\left({\theta }_3-1\right),\delta +1,2{\theta }_1\right),\hfill & \mathrm{if}\delta =1\text{;}\hfill \\ \left(2\left({\theta }_3-1\right),\delta +1,2{\theta }_1\right),\hfill & \mathrm{if}\delta =2\text{;}\hfill \\ \left(2\left({\theta }_3-2\right)+\delta ,\delta +1,2{\theta }_1-3+\delta \right),\hfill & \mathrm{if}\delta =3,4,\dots ,2{\theta }_2-1\text{.}\hfill \end{array}\right.$$

Positions $p\left(a_{\delta }{a}_{\delta }^{\prime }|F_m\right)$ with regard to $F_m$ , for the bonds $a_{\delta }{a}_{\delta }^{\prime }$ with $\delta =1,3,\dots ,2{\theta }_3-3$ , are described as: $$p\left(a_{\delta }{a}_{\delta }^{\prime }|F_m\right)=\left(\delta -1,2{\theta }_3-\delta ,2\left({\theta }_3+{\theta }_1\right)-1-\delta \right)\text{.}$$

Positions $p\left(b_{\delta +3}{b}_{\delta }^{\prime }|F_m\right)$ with regard to $F_m$ , for the bonds $b_{\delta +3}{b}_{\delta }^{\prime }$ with $\delta =1,3,\dots ,2{\theta }_1-3$ , are described as: $$p\left(b_{\delta +3}{b}_{\delta }^{\prime }|F_m\right)=\left(2{\theta }_3+\delta +1,\delta +3,2{\theta }_1-3-\delta \right)\text{.}$$

Positions $p\left(c_{\delta }{a}_{\delta }^{\prime }|F_m\right)$ with regard to $F_m$ , for the bonds $c_{\delta }{c}_{\delta }^{\prime }$ with $\delta =1,3,\dots ,2{\theta }_2-3$ , are described as: $$p\left(c_{\delta }{c}_{\delta }^{\prime }|F_m\right)=\left\{\begin{array}{cc}\left(2{\theta }_3-1,\delta +1,2{\theta }_1-1\right),\hfill & \mathrm{if}\delta =1\text{;}\hfill \\ \left(2\left({\theta }_3-2\right)+\delta ,\delta +1,2\left({\theta }_1-2\right)+\delta \right),\hfill & \mathrm{if}\delta =2,3,\dots ,2{\theta }_2-1\text{.}\hfill \end{array}\right.$$

Positions of the joint edges are described as: $$\begin{array}{c}\hfill p\left(a_{2{\theta }_3}{b}_1|F_m\right)=\left(2{\theta }_3-1,0,2{\theta }_1-1\right),\\ \hfill p\left(a_{2{\theta }_3}{c}_1^{\prime }|F_m\right)=\left(2{\theta }_3-2,1,2{\theta }_1\right),\\ \hfill p\left(b_2{c}_1|F_m\right)=\left(2{\theta }_3,2,2{\theta }_1-2\right),\\ \hfill p\left(a_{2{\theta }_3-3}^{\prime }{c}_2^{\prime }|F_m\right)=\left(2{\theta }_3-3,3,2{\theta }_1+1\right),\\ \hfill p\left(b_1^{\prime }{c}_2|F_m\right)=\left(2{\theta }_3+1,4,2{\theta }_1-3\right)\text{.}\end{array}$$

By the given positions $p\left(.|F_m\right)$ of all $4\left({\theta }_1+{\theta }_2+{\theta }_3\right)-8$ -nodes and $5\left({\theta }_1+{\theta }_2+{\theta }_3\right)-11$ -edges of $T\left({\theta }_1,{\theta }_2,{\theta }_3\right)$ graph of benzenoid tripod with any of ${\theta }_1,{\theta }_2,{\theta }_3=2$ , with respect to $F_m$ , are unique and no two nodes, no two edges and not a single edge with a vertex have same position $p$ . As a result, we may infer that the nodes and edges of the $T\left({\theta }_1,{\theta }_2,{\theta }_3\right)$ graph have been resolved with only three nodes. So it is concluded that the members with minimality conditions of mixed metric resolving set of $T\left({\theta }_1,{\theta }_2,{\theta }_3\right)$ are three. □

If $T\left({\theta }_1,{\theta }_2,{\theta }_3\right)$ is a of benzenoid tripod graph with any of ${\theta }_1,{\theta }_2,{\theta }_3=2$ , then $${\mathit{dim}}_m\left(T\left({\theta }_1,{\theta }_2,{\theta }_3\right)\right)=3\text{.}$$

As based on the definition of mixed metric dimension, the concept is solemnly based on the selected subset ( $F_m$ ) chooses in a manner that each edge and vertex have distinct location in regards to the chosen vertices in subset. The Lemma 2.1 shows and proved the chosen subset for the mixed metric resolving set and their hardness to be chosen with the condition of minimality. We proved in that lemma the subset $F_m=\{a_1,a_{2{\theta }_3},b_{2{\theta }_1}\}$ is one of the candidate for the mixed metric resolving set for a particular benzenoid tripod or $T\left({\theta }_1,{\theta }_2,{\theta }_3\right)$ graph for all the possible combinatorial values of with any of ${\theta }_1,{\theta }_2,{\theta }_3=2$ . Moreover, we discussed in the same lemma that $\left|F_m\right|=3$ is fulfilling the minimality conditions on their members. It is impossible to get a subset with two members in a selected subset. It is sufficient to verify what we assert in the assertion that the mixed metric dimension of the benzenoid tripod is three, which concludes the proof. □

If $T\left({\theta }_1,{\theta }_2,{\theta }_3\right)$ is a benzenoid tripod graph with ${\theta }_1,{\theta }_2,{\theta }_3⩾3$ , then the mixed metric resolving set has minimum four members.

The order or the total amount of vertices in the benzenoid tripod’s associated graph with any of ${\theta }_1,{\theta }_2,{\theta }_3⩾3$ , are $4\left({\theta }_1+{\theta }_2+{\theta }_3\right)-8$ , and to look over the likelihood of mixed metric resolvability set having four members in it, are $C\left(4\left({\theta }_1+{\theta }_2+{\theta }_3\right)-8,4\right)=\frac{\left(4\left({\theta }_1+{\theta }_2+{\theta }_3\right)-8\right)!}{2\times \left(4\left({\theta }_1+{\theta }_2+{\theta }_3\right)-12\right)!}$ . Over here we are focusing on four members in the set, for the three cardinality of mixed resolving set we will discuss in next part of this proof. Now because of its computational cost of selecting mixed metric resolving set, for a graph, we can’t discover the precise number of mixed resolving sets, but from $\frac{\left(4\left({\theta }_1+{\theta }_2+{\theta }_3\right)-8\right)!}{2\times \left(4\left({\theta }_1+{\theta }_2+{\theta }_3\right)-12\right)!}$ -possibilities we selected the subset $F_m$ and described as: $F_m=\{a_1,a_{2{\theta }_3},b_{2{\theta }_1},c_{2{\theta }_2-1}\}$ . Proving of this claim that $F_m$ is also a subset having ability to be the candidate of mixed metric resolving set of benzenoid tripod graph or $T\left({\theta }_1,{\theta }_2,{\theta }_3\right)$ , we will stick to 1.2’s definition. We will also look over the distinct placements or locations of each node and edge to meet the definition’s requirements, and the approach is explained above in Definition 1.2.

Positions $p\left(a_{\delta }|F_m\right)$ with respect to $F_m$ , for the nodes $a_{\delta }$ with $\delta =1,2,\dots ,2{\theta }_3$ , are described as: $$p\left(a_{\delta }|F_m\right)=\left\{\begin{array}{cc}\left(\delta -1,2{\theta }_3-\delta ,2\left({\theta }_3+{\theta }_1\right)-\delta ,2\left({\theta }_3+{\theta }_2\right)-3-\delta \right),\hfill & \mathrm{if}\delta =1,2,\dots ,2{\theta }_3-3\text{;}\hfill \\ \left(\delta -1,2{\theta }_3-\delta ,2\left({\theta }_3+{\theta }_1\right)-\delta ,2{\theta }_2+1\right),\hfill & \mathrm{if}\delta =2{\theta }_3-2\text{;}\hfill \\ \left(\delta -1,2{\theta }_3-\delta ,2\left({\theta }_3+{\theta }_1\right)-\delta ,2{\theta }_2\right),\hfill & \mathrm{if}\delta =2{\theta }_3-1\text{;}\hfill \\ \left(\delta -1,2{\theta }_3-\delta ,2\left({\theta }_3+{\theta }_1\right)-\delta ,2{\theta }_2+1\right),\hfill & \mathrm{if}\delta =2{\theta }_3\text{.}\hfill \end{array}\right.$$

Positions $p\left(b_{\delta }|F_m\right)$ with respect to $F_m$ , for the nodes $b_{\delta }$ with $\delta =1,2,\dots ,2{\theta }_1$ , are described as: $$p\left(b_{\delta }|F_m\right)=\left\{\begin{array}{cc}\left(2{\theta }_3+\delta -1,\delta ,2{\theta }_1-\delta ,2{\theta }_2+1-\delta \right),\hfill & \mathrm{if}\delta =1,2\text{;}\hfill \\ \left(2{\theta }_3+\delta -1,\delta ,2{\theta }_1-\delta ,2{\theta }_2\right),\hfill & \mathrm{if}\delta =3\text{;}\hfill \\ \left(2{\theta }_3+\delta -1,\delta ,2{\theta }_1-\delta ,2{\theta }_2-5+\delta \right),\hfill & \mathrm{if}\delta =4,5,\dots ,2{\theta }_1\text{.}\hfill \end{array}\right.$$

Positions $p\left(c_{\delta }|F_m\right)$ with respect to $F_m$ , for the nodes $c_{\delta }$ with $\delta =1,2,\dots ,2{\theta }_2-1$ , are described as: $$p\left(c_{\delta }|F_m\right)=\left\{\begin{array}{cc}\left(2{\theta }_3+\delta -1,\delta +2,2{\theta }_1-1,2{\theta }_2-\delta -1\right),\hfill & \mathrm{if}\delta =1\text{;}\hfill \\ \left(2{\theta }_3+\delta -1,\delta +2,2\left({\theta }_1-2\right)+\delta ,2{\theta }_2-\delta -1\right),\hfill & \mathrm{if}\delta =2\text{;}\hfill \\ \left(2{\theta }_3+\delta -3,\delta +2,2\left({\theta }_1-2\right)+\delta ,2{\theta }_2-\delta -1\right),\hfill & \mathrm{if}\delta =3,4,\dots ,2{\theta }_2-1\text{.}\hfill \end{array}\right.$$

Positions $p\left(a_{\delta }^{\prime }|F_m\right)$ with respect to $F_m$ , for the nodes $a_{\delta }^{\prime }$ with $\delta =1,2,\dots ,2{\theta }_3-3$ , are described as: $$p\left(a_{\delta }^{\prime }|F_m\right)=\left(\delta ,2{\theta }_3+1-\delta ,2\left({\theta }_3+{\theta }_1\right)-1-\delta ,2\left({\theta }_3+{\theta }_1-2\right)-\delta \right)\text{.}$$

Positions $p\left(b_{\delta }^{\prime }|F_m\right)$ with respect to $F_m$ , for the nodes $b_{\delta }^{\prime }$ with $\delta =1,2,\dots ,2{\theta }_1-3$ , are described as: $$p\left(b_{\delta }^{\prime }|F_m\right)=\left(2{\theta }_3+\delta +1,\delta +4,2\left({\theta }_1-1\right)-\delta ,2{\theta }_2-3+\delta \right)\text{.}$$

Positions $p\left(c_{\delta }^{\prime }|F_m\right)$ with respect to $F_m$ , for the nodes $c_{\delta }^{\prime }$ with $\delta =1,2,\dots ,2{\theta }_2-1$ , are described as: $$p\left(c_{\delta }^{\prime }|F_m\right)=\left\{\begin{array}{cc}\left(2{\theta }_3-1,\delta +1,2{\theta }_1-1+\delta ,2{\theta }_2-\delta \right),\hfill & \mathrm{if}\delta =1\text{;}\hfill \\ \left(2\left({\theta }_3-2\right)+\delta ,\delta +1,2{\theta }_1-1+\delta ,2{\theta }_2-\delta \right),\hfill & \mathrm{if}\delta =2\text{;}\hfill \\ \left(2\left({\theta }_3-2\right)+\delta ,\delta +1,2{\theta }_1-3+\delta ,2{\theta }_2-\delta \right),\hfill & \mathrm{if}\delta =3,4,\dots ,2{\theta }_2-1\text{.}\hfill \end{array}\right.$$

Given above are the positions of nodes only, with respect to the chosen vertices in $F_m$ , for the fulfillment of definition of mixed resolving set we still need to investigates the edges or bonds of graph. Following are positions of bonds that will complete the definition.

Positions $p\left(a_{\delta }{a}_{\delta +1}|F_m\right)$ with respect to $F_m$ , for the bonds $a_{\delta }{a}_{\delta +1}$ with $\delta =1,2,\dots ,2{\theta }_3-1$ , are described as: $$p\left(a_{\delta }{a}_{\delta +1}|F_m\right)=\left\{\begin{array}{c}\left(\delta -1,2{\theta }_3-1-\delta ,2\left({\theta }_3+{\theta }_1\right)-1-\delta ,2\left({\theta }_3+{\theta }_2-2\right)-\delta \right),\hfill \\ \mathrm{if}\delta =1,2,\dots ,2{\theta }_3-4\text{;}\hfill \\ \left(\delta -1,2{\theta }_3-1-\delta ,2\left({\theta }_3+{\theta }_1\right)-1-\delta ,2{\theta }_2\right),\hfill \\ \mathrm{if}\delta =2{\theta }_3-3,2{\theta }_3-2\text{;}\hfill \\ \left(\delta -1,2{\theta }_3-1-\delta ,2{\theta }_3,2{\theta }_2\right),\hfill \\ \mathrm{if}\delta =2{\theta }_3-1\text{.}\hfill \end{array}\right.$$

Positions $p\left(b_{\delta }{b}_{\delta +1}|F_m\right)$ in regards to $F_m$ , for the bonds $b_{\delta }{b}_{\delta +1}$ with $\delta =1,2,\dots ,2{\theta }_1-1$ , are described as: $$p\left(b_{\delta }{b}_{\delta +1}|F_m\right)=\left\{\begin{array}{cc}\left(2{\theta }_3+\delta -1,\delta ,2{\theta }_1-1-\delta ,2{\theta }_2-1\right),\hfill & \mathrm{if}\delta =1,2,3\text{;}\hfill \\ \left(2{\theta }_3+\delta -1,\delta ,2{\theta }_1-1-\delta ,2{\theta }_2-5+\delta \right),\hfill & \mathrm{if}\delta =4,5,\dots ,2{\theta }_1-1\text{.}\hfill \end{array}\right.$$

Positions $p\left(c_{\delta }{c}_{\delta +1}|F_m\right)$ in regards to $F_m$ , for the bonds $c_{\delta }{c}_{\delta +1}$ with $\delta =1,2,\dots ,2{\theta }_2-2$ , are described as: $$p\left(c_{\delta }{c}_{\delta +1}|F_m\right)=\left\{\begin{array}{cc}\left(2{\theta }_3,\delta +2,2{\theta }_1-2,2{\theta }_2-2-\delta \right),\hfill & \mathrm{if}\delta =1\text{;}\hfill \\ \left(2{\theta }_3,\delta +2,2\left({\theta }_1-2\right)+\delta ,2{\theta }_2-2-\delta \right),\hfill & \mathrm{if}\delta =2\text{;}\hfill \\ \left(2{\theta }_3+\delta -3,\delta +2,2\left({\theta }_1-2\right)+\delta ,2{\theta }_2-2-\delta \right),\hfill & \mathrm{if}\delta =3,4,\dots ,2{\theta }_2-1\text{.}\hfill \end{array}\right.$$

Positions $p\left(a_{\delta }^{\prime }{a}_{\delta +1}^{\prime }|F_m\right)$ in regards to $F_m$ , for the bonds $a_{\delta }^{\prime }{a}_{\delta +1}^{\prime }$ with $\delta =1,2,\dots ,2{\theta }_3-4$ , are described as: $$p\left(a_{\delta }^{\prime }{a}_{\delta +1}^{\prime }|F_m\right)=\left(\delta ,2{\theta }_3-\delta ,2\left({\theta }_3+{\theta }_1-1\right)-\delta ,2\left({\theta }_3+{\theta }_1\right)-5-\delta \right)\text{.}$$

Positions $p\left(b_{\delta }^{\prime }{b}_{\delta +1}^{\prime }|F_m\right)$ in regards to $F_m$ , for the bonds $b_{\delta }^{\prime }{b}_{\delta +1}^{\prime }$ with $\delta =1,2,\dots ,2{\theta }_1-4$ , are described as: $$p\left(b_{\delta }^{\prime }{b}_{\delta +1}^{\prime }|F_m\right)=\left(2{\theta }_3+\delta +1,\delta +4,2{\theta }_1-3-\delta ,2{\theta }_2-3+\delta \right)\text{.}$$

Positions $p\left(c_{\delta }^{\prime }{c}_{\delta +1}^{\prime }|F_m\right)$ in regards to $F_m$ , for the bonds $c_{\delta }^{\prime }{c}_{\delta +1}^{\prime }$ with $\delta =1,2,\dots ,2{\theta }_2-2$ , are described as: $$p\left(c_{\delta }^{\prime }{c}_{\delta +1}^{\prime }|F_m\right)=\left\{\begin{array}{cc}\left(2\left({\theta }_3-1\right),\delta +1,2{\theta }_1,2{\theta }_2-3+\delta \right),\hfill & \mathrm{if}\delta =1\text{;}\hfill \\ \left(2\left({\theta }_3-1\right),\delta +1,2{\theta }_1,2{\theta }_2-3+\delta \right),\hfill & \mathrm{if}\delta =2\text{;}\hfill \\ \left(2\left({\theta }_3-2\right)+\delta ,\delta +1,2{\theta }_1-3+\delta ,2{\theta }_2-3+\delta \right),\hfill & \mathrm{if}\delta =3,4,\dots ,2{\theta }_2-1\text{.}\hfill \end{array}\right.$$

Positions $p\left(a_{\delta }{a}_{\delta }^{\prime }|F_m\right)$ in regards to $F_m$ , for the bonds $a_{\delta }{a}_{\delta }^{\prime }$ with $\delta =1,3,\dots ,2{\theta }_3-3$ , are described as: $$p\left(a_{\delta }{a}_{\delta }^{\prime }|F_m\right)=\left(\delta -1,2{\theta }_3-\delta ,2\left({\theta }_3+{\theta }_1\right)-1-\delta ,2\left({\theta }_3+{\theta }_2-2\right)-\delta \right)\text{.}$$

Positions $p\left(b_{\delta +3}{b}_{\delta }^{\prime }|F_m\right)$ in regards to $F_m$ , for the bonds $b_{\delta +3}{b}_{\delta }^{\prime }$ with $\delta =1,3,\dots ,2{\theta }_1-3$ , are described as: $$p\left(b_{\delta +3}{b}_{\delta }^{\prime }|F_m\right)=\left(2{\theta }_3+\delta +1,\delta +3,2{\theta }_1-3-\delta ,2{\theta }_2-3+\delta \right)\text{.}$$