# Analytical Treatment of Higher-Order Graphs: A Path Ordinal Method for Solving Graphs

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Flow Graphs

#### 2.2. Graph Composition and Path Characterization

_{m,n}= m

^{n+1}.

^{2}. This individual graph is defined by the incoming nodes vector ${\overrightarrow{x}}_{j-1}$ and the outgoing nodes vector ${\overrightarrow{x}}_{j}$, where j takes the values j = 1, 2, 3, …, n.

#### 2.2.1. Ordinal of a Path and Path Value

_{m,n}), connecting any node in the input vector with another node in the output vector, is characterized by a set of numbers {θ

_{ij}} that we will call “path-set”, which defines the trajectory of the path.

_{ij}can take any of the values 1 ≤ θ

_{ij}≤ m. If θ

_{ij}= k this means that the ith path passes through the kth node of the jth vector, x

_{jk}. An example of a path-set is illustrated in Figure 6.

_{i}can be seen as the product of the transmittances corresponding to each branch along the path

_{ij}is the transmittance of the branch in the jth graph within the path i (see Figure 5).

_{i}that starts in an arbitrary node x

_{0k}in the input vector ${\overrightarrow{x}}_{0}$ and reaches the output vector ${\overrightarrow{x}}_{n}$ in any arbitrary node x

_{nL}, is given by

_{01}and will end at the node x

_{n}

_{1}, the second starts from the node x

_{02}and ends at the node x

_{n}

_{1}, the kth will start from the node x

_{0k}till the path m is reached, which starts from the node x

_{0m}and ends at the node x

_{n}

_{1}.

_{m,n}is divided into m groups each has m

^{n}paths. The first group ends at the node x

_{1n}where (1 ≤ i ≤ m

^{n}) and the second group ends at the node x

_{n}

_{2}where (1 + m

^{n}≤ i ≤ 2m

^{n}). As a consequence, all of the paths that end at the node x

_{nL}have the path ordinals within the limit (1 + (L − 1)m

^{n}≤ i ≤ Lm

^{n}). On the other hand, all of the paths that start from the node x

_{0k}, according to the path sequence, have the ordinals i = k, k + m, k + 2m, ….

^{n−}

^{1}paths connecting an output node with an input one. These paths that start from an input node x

_{0k}and ends at an output node x

_{nL}, have the path ordinals i = k + (L − 1)m

^{n}, k + m+ (L − 1)m

^{n}, …, k – m + Lm

^{n}.

_{0k}to the sink x

_{nL}, can be expressed as the summation of all the paths that start from x

_{0k}and end at x

_{nL}as follows,

_{nL}is given by,

^{kL}to the summation of all the path values that start from x

_{0k}and end at x

_{nL}

_{nL}can be expressd as,

_{i}

_{0}and θ

_{in}are defined, with the values k and L, respectively. Now, the goal is to define the trajectory of an arbitrary path, i.e., to evaluate the set of numbers {θ

_{ij}}.

#### 2.2.2. Determination of the Characteristic Path Set

^{n}paths. Each group reaches an output node. Accordingly, the group of paths that reaches an arbitrary node x

_{nL}in the output vector ${\overrightarrow{x}}_{n}$ has the path ordinals within the range (1 + (L − 1)m

^{n}≤ i ≤ Lm

^{n}).

^{n}), we get:

^{n}is de divisor, C

_{n}is the quotient (0 ≤ C

_{n}< m), and R

_{n}is the reminder (0 ≤ R

_{n}< m

^{n}). So, the last element of the path-set θ

_{in}can be expressed as:

^{n}

^{−1}paths. Now, the path ordinal becomes:

_{n}+ 1.

_{(n−1)L}that the path ends at, following the previous procedure, the path ordinal is subtracted by one and then divided by m

^{n−}

^{1}, so we have:

_{n}

_{−1}< m and 0 ≤ R

_{n}

_{−1}< m

_{n}

_{−1}. Iterating the same procedure, we finally get:

_{ij}} can be determined as follows:

- I
- The path ordinal is subtracted by one.
- II
- Then, it is divided by m for n-times.
- III
- Finally, one is added to the remainders of the division, R
_{1}, C_{1}, C_{2}, …, C_{n}.

#### 2.3. Examples and Concluding Remarks

#### 2.3.1. The Contribution of and Input Parameter to an Output Parameter

_{03}to the output parameter x

_{22}is given by the matrix element T

^{32}. When applying Equation (8), we get:

12 − 1 = | 11 | 3 | 15 − 1 = | 14 | 3 | 18 − 1 = | 17 | 3 | |||||

: | 3 | 3 | : | 4 | 3 | : | 5 | 3 | |||||

: | : | 1 | : | : | 1 | : | : | 1 | |||||

: | : | : | : | : | : | : | : | : | |||||

2 | 0 | 1 | 2 | 1 | 1 | 2 | 2 | 1 | |||||

+1 | 3 | 1 | 2 | +1 | 3 | 2 | 2 | +1 | 3 | 3 | 2 |

#### 2.3.2. The Total Solution

## 3. Results and Discussion

_{m,n}), which, as a consequence, is attached to a characteristic Path-Set that determines the path along the graph and a Path-Value that is considered as the product of the transmittances corresponding to each branch along the path. Once the path values are calculated, the transmittances of the branches of the residual graph are calculated through Equation (8), which are homologues to the matrix elements representing the system. For better organization, simplicity, and in order to avoid calculation mistakes, we suggest that all of the calculations to be put in tables. Table 4 and Table 5, and Figure 10 summarizes the process.

_{0k}to a sink x

_{nL}is given by the transmittance T

^{kL}of the residual graph. Which is the summation of its corresponding path values in agreement with Equation (8). These path values can be calculated directly by specifying their corresponding path sets by means of the PSD illustrated in Figure 7, i.e., by applying the POM the contribution of a source to a sink can be calculated easily without solving the entire problem.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 2.**A second order graph representing Equation (1). The homologous of the vector parameters are the nodes while the homologous of the matrix elements are the branches transmittance.

**Figure 3.**A cascade graph composed of n graphs of order two, attached side by side. Each individual graph represents a 2 × 2 matrix.

**Figure 5.**The jth graph of order m inside the cascade graph of Figure 4. A

_{j}(k,L) represents the transmittance of the branch connecting the nodes x

_{k}and x

_{L}.

**Figure 6.**An example illustrating a path i. The trajectory of the path corresponds to the path-set {θ

_{ij}} = {2,3,1,…,k,L,…,m − 1,m} where x

_{mn}= P

_{i}x

_{20}and P

_{i}= (A

_{1}(2,3)·A

_{1}(3,1)·…A

_{j}(k,L)·…A

_{n}(m − 1,m)).

**Figure 7.**The Path Set Diagram (PSD). Starting from a path ordinal 𝑖, its corresponding characteristic path set $\left\{{\theta}_{ij}\right\}=\left\{{\theta}_{i0},{\theta}_{i1},{\theta}_{i2},{\theta}_{i3},\dots \dots \dots ,{\theta}_{in}\right\}$ is calculated as follows: firstly, the path ordinal is subtracted by one, then it is divided by m n-times, finally one is added to the remainders of the division.

**Figure 9.**The three possible graph paths P

_{12}, P

_{15}, and P

_{18}, connecting the input node x

_{03}to the output node x

_{22}according to the path sets calculated by the Path Set Diagram (PSD).

**Table 1.**The path-sets corresponding to the 3 × 2 cascade graph. The table is divided vertically into three parts, each part represents the paths that reach the output nodes x

_{21}, x

_{22}and x

_{23}, respectively.

i | θ_{i0} | θ_{i1} | θ_{i2} | i | θ_{i0} | θ_{i1} | θ_{i2} | i | θ_{i0} | θ_{i1} | θ_{i2} | ||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 1 | 1 | 1 | 10 | 1 | 1 | 2 | 19 | 1 | 1 | 3 | ||

2 | 2 | 1 | 1 | 11 | 2 | 1 | 2 | 20 | 2 | 1 | 3 | ||

3 | 3 | 1 | 1 | 12 | 3 | 1 | 2 | 21 | 3 | 1 | 3 | ||

4 | 1 | 2 | 1 | 13 | 1 | 2 | 2 | 22 | 1 | 2 | 3 | ||

5 | 2 | 2 | 1 | 14 | 2 | 2 | 2 | 23 | 2 | 2 | 3 | ||

6 | 3 | 2 | 1 | 15 | 3 | 2 | 2 | 24 | 3 | 2 | 3 | ||

7 | 1 | 3 | 1 | 16 | 1 | 3 | 2 | 25 | 1 | 3 | 3 | ||

8 | 2 | 3 | 1 | 17 | 2 | 3 | 2 | 26 | 2 | 3 | 3 | ||

9 | 3 | 3 | 1 | 18 | 3 | 3 | 2 | 27 | 3 | 3 | 3 |

**Table 2.**The path values corresponding to the path-sets of Table 1.

P_{i} | Path Value | P_{i} | Path Value | P_{i} | Path Value |
---|---|---|---|---|---|

P_{1} | A_{1}(1,1)*A_{2}(1,1) | P_{10} | A_{1}(1,1)*A_{2}(1,2) | P_{19} | A_{1}(1,1)*A_{2}(1,3) |

P_{2} | A_{1}(2,1)*A_{2}(1,1) | P_{11} | A_{1}(2,1)*A_{2}(1,2) | P_{20} | A_{1}(2,1)*A_{2}(1,3) |

P_{3} | A_{1}(3,1)*A_{2}(1,1) | P_{12} | A_{1}(3,1)*A_{2}(1,2) | P_{21} | A_{1}(3,1)*A_{2}(1,3) |

P_{4} | A_{1}(1,2)*A_{2}(2,1) | P_{13} | A_{1}(1,2)*A_{2}(2,2) | P_{22} | A_{1}(1,2)*A_{2}(2,3) |

P_{5} | A_{1}(2,2)*A_{2}(2,1) | P_{14} | A_{1}(2,2)*A_{2}(2,2) | P_{23} | A_{1}(2,2)*A_{2}(2,3) |

P_{6} | A_{1}(3,2)*A_{2}(2,1) | P_{15} | A_{1}(3,2)*A_{2}(2,2) | P_{24} | A_{1}(3,2)*A_{2}(2,3) |

P_{7} | A_{1}(1,3)*A_{2}(3,1) | P_{16} | A_{1}(1,3)*A_{2}(3,2) | P_{25} | A_{1}(1,3)*A_{2}(3,3) |

P_{8} | A_{1}(2,3)*A_{2}(3,1) | P_{17} | A_{1}(2,3)*A_{2}(3,2) | P_{26} | A_{1}(2,3)*A_{2}(3,3) |

P_{9} | A_{1}(3,3)*A_{2}(3,1) | P_{18} | A_{1}(3,3)*A_{2}(3,2) | P_{27} | A_{1}(3,3)*A_{2}(3,3) |

T^{11} = P_{1} + P_{4} + P_{7} | T^{21} = P_{2} + P_{5} + P_{8} | T^{31} = P_{3} + P_{6} + P_{9} |

T^{12} = P_{10} + P_{13} + P_{16} | T^{22} = P_{11} + P_{14} + P_{17} | T^{32} = P_{12} + P_{15} + P_{18} |

T^{13} = P_{19} + P_{22} + P_{25} | T^{23} = P_{20} + P_{23} + P_{26} | T^{33} = P_{21} + P_{24} + P_{27} |

**Table 4.**A general form of a table used to calculate all the possible paths of a $m\times n$ cascade graph. Aj(k,L) represents the transmittance of the branch connecting the node X(j − 1)k and Xjk in the jth graph.

Path Ordinal P_{i} | Path Set $\left\{{\mathit{\theta}}_{\mathit{i}\mathit{j}}\right\}=\left\{{\mathit{\theta}}_{\mathit{i}0},{\mathit{\theta}}_{\mathit{i}1},{\mathit{\theta}}_{\mathit{i}2},..........,{\mathit{\theta}}_{\mathit{i}\mathit{n}}\right\}$ | Path Value ${\mathit{P}}_{\mathit{i}}={\displaystyle \prod _{\mathit{j}=1}^{\mathit{n}}{\mathit{A}}_{\mathit{i}\mathit{j}}}({\mathit{\theta}}_{\mathit{i}(\mathit{j}-1)},{\mathit{\theta}}_{\mathit{i}\mathit{j}})$ |
---|---|---|

1 | {1, 1…………...…..., 1} | P_{1} = A_{1}(1,1)*………………A_{n}(1,1) |

: | : | : |

m^{n} | {m, m…………….…, 1} | ${P}_{{m}^{n}}$ = A_{1}(m,m)*………….A_{n}(m,1) |

: | : | : |

: | : | : |

m^{n+}^{1} | {m, m……………..., m} | ${P}_{{m}^{n+1}}$ = A_{1}(m,m)*………...A_{n}(m,m) |

**Table 5.**A table illustrating the value of each transmittance in the residual graph as a sum of its corresponding path-values.

${T}^{11}={\displaystyle {\displaystyle \sum}_{r=1}^{{m}^{n-1}}}{P}_{1-m+r.m}^{}$ | … | ${T}^{k1}={\displaystyle {\displaystyle \sum}_{r=1}^{{m}^{n-1}}}{P}_{k-m+r.m}^{}$ | … | ${T}^{m1}={\displaystyle {\displaystyle \sum}_{r=1}^{{m}^{n-1}}}{P}_{r.m}^{}$ |

: : | … | : : | … | : : |

${T}^{1L}={\displaystyle {\displaystyle \sum}_{r=1}^{{m}^{n-1}}}{P}_{1-m+\left(L-1\right).{m}^{n}+r.m}$ | … | ${T}^{kL}={\displaystyle {\displaystyle \sum}_{r=1}^{{m}^{n-1}}}{P}_{k-m+\left(L-1\right).{m}^{n}+r.m}$ | … | ${T}^{mL}={\displaystyle {\displaystyle \sum}_{r=1}^{{m}^{n-1}}}{P}_{\left(L-1\right).{m}^{n}+r.m}$ |

: : | … | : : | … | : : |

${T}^{1m}={\displaystyle {\displaystyle \sum}_{r=1}^{{m}^{n-1}}}{P}_{1-m+\left(m-1\right).{m}^{n}+r.m}$ | … | ${T}^{km}={\displaystyle {\displaystyle \sum}_{r=1}^{{m}^{n-1}}}{P}_{k-m+\left(m-1\right).{m}^{n}+r.m}$ | … | ${T}^{mm}={\displaystyle {\displaystyle \sum}_{r=1}^{{m}^{n-1}}}{P}_{\left(m-1\right).{m}^{n}+r.m}$ |

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**MDPI and ACS Style**

Kamal, H.; Larena, A.; Bernabeu, E.
Analytical Treatment of Higher-Order Graphs: A Path Ordinal Method for Solving Graphs. *Symmetry* **2017**, *9*, 288.
https://doi.org/10.3390/sym9110288

**AMA Style**

Kamal H, Larena A, Bernabeu E.
Analytical Treatment of Higher-Order Graphs: A Path Ordinal Method for Solving Graphs. *Symmetry*. 2017; 9(11):288.
https://doi.org/10.3390/sym9110288

**Chicago/Turabian Style**

Kamal, Hala, Alicia Larena, and Eusebio Bernabeu.
2017. "Analytical Treatment of Higher-Order Graphs: A Path Ordinal Method for Solving Graphs" *Symmetry* 9, no. 11: 288.
https://doi.org/10.3390/sym9110288