Figures of Graph Partitioning by Counting, Sequence and Layer Matrices
Abstract
:1. Introduction
Related Research
2. Graphs and Their Representation
3. Counting Matrices
4. Collecting Sets of Vertices
5. Layer Matrices
6. Sequence Matrices
7. Paths and Cycles
8. Distinct Partitions Coloring of Vertices
9. Case Study for Isomers of ${\mathit{C}}_{\mathbf{28}}$ Fullerene
10. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Sample Availability
Appendix A. Molecular Graphs and Their Representation
 First three lines—reserved for compound identification
 On fourth line: the number of atoms, the number of bonds, followed by a series of (8) reserved values (numeric and string)
 An block of lines describing on each line one atom: the cartezian coordinates (x, y, and z), the symbol of the atom and a series of (12) reserved fields (numeric)
 An block of lines describing on each line one bond: two numbers acting as indices for the atoms and a third number indicating the bond order, followed by a series of (4) reserved fields (numeric)
 First two lines—reserved for compound identification
 An block of lines describing on each line one atom: type of the fragment, index (numeric), symbol of the atom (1–3 characters), two other columns followed by the cartezian coordinates (x, y, and z)
 An block of lines describing on each line the topology for one atom: atom index followed by the indices of the atoms connected with it
 in between mol $number$ and endmol $number$ on each line one atom having on the second column the atom index, on the fourth the atom symbol, from column 8 to 11 the cartesian coordinates, on column 12 the number of bonds followed (starting with column 13) by each bond on two columns each (atom index, bond order)
Appendix B. Algorithms
$\mathit{tv}$  0  1  …  

1  ${c}_{1,0}$  ${c}_{1,1}$  …  ${c}_{1,{c}_{1,0}}$ 
…  …  …  …  … 
n  ${c}_{n,0}$  ${c}_{n,1}$  …  ${c}_{n,{c}_{n,0}}$ 
Algorithm A1 Set adjacency matrix. 
Output:a //a→Ad, Ad—the adjacency matrix ($\mathrm{n}\times \mathrm{n}$) 
Algorithm A2 Set distance matrix. 
Input:n, a //n←n, a←Ad
Output:d //d→Di, Di—the distance matrix ($\mathrm{n}\times \mathrm{n}$) 
Algorithm A3 Set graph diameter. 
Input:n, d //n←n, d←Di
Output:e //e→d, d—the diameter (longest distance) 
Algorithm A4 Set LD2 matrix. 
Input:n, e, f, g //n←n, e←d, f←LD0, g←Dic
Output:h //h→LD2, the LD2 matrix 
Algorithm A5 Set LD3 matrix. 
Input:n, e, $vs.$, a, f //n←n, v←tv, a←Ad, e←d, f←LD0
Output:r //r→LD3, the LD3 matrix 
Algorithm A6 Set LD4 matrix. 
Input:n, e, b, c //n←n, e←d; b←LD2, c←LD3
Output:s //s→LD4, the LD4 matrix 
Algorithm A7 Set LD5 matrix. 
Input:n, e, d, f //n←n, a←Ad, e←d, g←LD0
Output:h //h→LD5, the LD5 matrix 
Algorithm A8 Set LkW matrix. 
Input:n, e, a, g //n←n, a←Ad, e←d, g←LD0
Output:h //h→LW, LW the L(1)W..L(e)W matrices 
Algorithm A9 Set Szeged matrix. 
Input:n, s //n←n, s←Szs
Output:d //d→Szd, Szd—the Szeged matrix 
Algorithm A10 Set layer sets. 
Input:n, e, m //n←n, e←d, m←Ad, Di,Szd (m—any vertex pair based matrix)
Output:s //s→LA0,LD0,LS0 (s—the layer set of the matrix m) 
Algorithm A11 Set counting matrix. 
Input:n, m //n←n, m←Ad,Di,Szd (any $n\times n$ matrix)
Output:c //c→Adc,Dic,Szc (counting matrix of m) 
Algorithm A12 Set distances of paths list. 
Input:n, c, d, e //n←n, c←tv; d←Di; e←d
Output:p //p→path, path—the distance paths 
Algorithm A13 Set cycles list. 
Input:n, a, e, p //n←n, a←Ad, e←d, p←path
Output:c //c→cycle, cycle—the ones no longer than the double of the diameter 
Algorithm A14 Set distance path sequence and layers. 
Input:n, e, p //n←n, e←d; p←path
Output:s, l //s→SP0, l→LP0—the distance paths sequence and layers 
Algorithm A15 Set cycle sequence and layers. 
Input:n, c //n←n, c←cycle
Output:s, l //s→SC0, l→LC0—the cycles sequence and layers 
Algorithm A16 Count and sum for paths and cycles. 
Input:n, $vs.$ //n←n; v←SP0,LP0,SC0, LC0
Output:s, c //c→SP1,LP1,SC1,LC1 (counts), s→SP2,LP2,SC2,LC2 (sums) 
Algorithm A17 Set Szeged sets. 
Input:n, d //n←n, d←Di
Output:s //s→Szs, Szs—the Szeged sets 
Algorithm A18 Use of the above algorithms. 

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$\left[\mathit{LA}0\right]$  0  1 

1  {1, 3, 5}  {2, 4} 
2  {2, 4, 5}  {1, 3} 
3  {1, 3, 5}  {2, 4} 
4  {2, 4}  {1, 3, 5} 
5  {1, 2, 3, 5}  {4} 
$\left[\mathit{LD}0\right]$  0  1  2  3 

1  {1}  {2, 4}  {3, 5}  {} 
2  {2}  {1, 3}  {4}  {5} 
3  {3}  {2, 4}  {1, 5}  {} 
4  {4}  {1, 3, 5}  {2}  {} 
5  {5}  {4}  {1, 3}  {2} 
$\left[\mathit{LS}0\right]$  0  1  2  3  4 

1  {1}  {3}  {4, 5}  {2}  {} 
2  {2}  {4}  {1, 3}  {5}  {} 
3  {3}  {1}  {4, 5}  {2}  {} 
4  {4}  {}  {2}  {1, 3}  {5} 
5  {5}  {1, 3, 4}  {2}  {}  {} 
$\left[\mathit{SP}0\right]$  1  2  3  4  5 

1  {}  {1 2}  {1 4 3, 1 2 3}  {1 4}  {1 4 5} 
2  {2 1}  {}  {2 3}  {2 3 4, 2 1 4}  {2 1 4 5, 2 3 4 5} 
3  {3 4 1, 3 2 1}  {3 2}  {}  {3 4}  {3 4 5} 
4  {4 1}  {4 3 2, 4 1 2}  {4 3}  {}  {4 5} 
5  {5 4 1}  {5 4 1 2, 5 4 3 2}  {5 4 3}  {5 4}  {} 
$\left[\mathit{LP}0\right]$  0  1  2  3 

1  {}  {1 2, 1 4}  {1 4 3, 1 4 5, 1 2 3}  {} 
2  {}  {2 1, 2 3}  {2 3 4, 2 1 4}  {2 1 4 5, 2 3 4 5} 
3  {}  {3 2, 3 4}  {3 4 5, 3 4 1, 3 2 1}  {} 
4  {}  {4 1, 4 3, 4 5}  {4 3 2, 4 1 2}  {} 
5  {}  {5 4}  {5 4 3, 5 4 1}  {5 4 1 2, 5 4 3 2} 
$\left[\mathit{SC}0\right]$  1  2  3  4  5 

1  {}  {1 2 3 4}  {1 2 3 4}  {1 2 3 4}  {} 
2  {1 2 3 4}  {}  {1 2 3 4}  {1 2 3 4}  {} 
3  {1 2 3 4}  {1 2 3 4}  {}  {1 2 3 4}  {} 
4  {1 2 3 4}  {1 2 3 4}  {1 2 3 4}  {}  {} 
5  {}  {}  {}  {}  {} 
$\left[\mathit{LC}0\right]$  3  4 

1  {}  {1 2 3 4} 
2  {}  {1 2 3 4} 
3  {}  {1 2 3 4} 
4  {}  {1 2 3 4} 
5  {}  {} 
$\left[\mathit{Szs}\right]$  1  2  3  4  5 

1  {}  {1, 4, 5}  {1}  {1, 2}  {1, 2} 
2  {2, 3}  {}  {1, 2}  {2}  {1, 2, 3} 
3  {3}  {3, 4, 5}  {}  {2, 3}  {2, 3} 
4  {3, 4, 5}  {4, 5}  {1, 4, 5}  {}  {1, 2, 3, 4} 
5  {5}  {4, 5}  {5}  {5}  {} 
Vertices  ${\mathit{\Sigma}}_{\mathit{j}}{\mathbf{Szd}}_{\mathit{i},\mathit{j}}$  “.”${}_{\mathit{k}}{\mathbf{Szc}}_{\mathit{i},\mathit{k}}$ 

1  8  1.1.2.1.0 
2  8  1.1.2.1.0 
3  8  1.1.2.1.0 
4  12  1.0.1.2.1 
5  5  1.3.1.0.0 
Order of the Vertices Sets Is Relevant  Classifier (from $\left[\mathit{Matrix}\right]$)  Order of the Vertices Sets Is Not Relevant 

{5}, {1, 2, 3}, {4}  Ad, Szd  {1, 2, 3}, {4}, {5} 
{4}, {1, 2, 3}, {5}  Adc, Szc  
{4}, {1, 3}, {2}, {5}  Di, LD5  {1, 3}, {2}, {4}, {5} 
{5}, {2}, {1, 3}, {4}  Dic, LD1, LD2, LD3, LD4, LW1, LW2, LP1, LP2  
{5}, {1, 3}, {2}, {4}  LW3  
{4}, {1,3}, {5}, {2}  SP2  
{1, 3, 4, 5}, {2}  SP1  {1, 3, 4, 5}, {2} 
{5}, {1, 2, 3, 4}  SC1, SC2, LC1, LC2  {1, 2, 3, 4}, {5} 
Order Is Relevant  Classifiers  Order Is Not Relevant 

D1  Ad, Adc  U1 
D2  Di  U2 
D4  Dic, LD1, LD2, LW1, LW2, LW3, LW4, LW5, LW6  
D3  Szd  U3 
D5  Szc  U4 
D6  LD3, LD4  
D7  LD5  
D11  SC1, SC2  
D12  LC1, LC2  
D8  SP1  U5 
D9  SP2  U6 
D10  LP1, LP2 
Order Is Relevant  Classifiers  Order Is Not Relevant 

D1  Ad, Adc  U1 
D2  Di  U2 
D3  Dic, LD1, LD2, LW1, LW2, LW3, LW4, LW5, LW6, Szd  U3 
D4  Szc, LD5, SP1, SP2, SC1, SC2, LC1, LC2  
D5  LD3, LD4, LP1, LP2 
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Tomescu, M.A.; Jäntschi, L.; Rotaru, D.I. Figures of Graph Partitioning by Counting, Sequence and Layer Matrices. Mathematics 2021, 9, 1419. https://doi.org/10.3390/math9121419
Tomescu MA, Jäntschi L, Rotaru DI. Figures of Graph Partitioning by Counting, Sequence and Layer Matrices. Mathematics. 2021; 9(12):1419. https://doi.org/10.3390/math9121419
Chicago/Turabian StyleTomescu, Mihaela Aurelia, Lorentz Jäntschi, and Doina Iulia Rotaru. 2021. "Figures of Graph Partitioning by Counting, Sequence and Layer Matrices" Mathematics 9, no. 12: 1419. https://doi.org/10.3390/math9121419