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Counting Paths, Circuits, Chains, and Cycles in Graphs: a Unified Approach

Published online by Cambridge University Press:  20 November 2018

E. Biondi ,
L. Divieti  and
G. Guardabassi
E. Biondi
Affiliation:
Institute di Elettrotecnica ed Elettronica del Politecnico di Milano, Milano, Italy
L. Divieti
Affiliation:
Institute di Elettrotecnica ed Elettronica del Politecnico di Milano, Milano, Italy
G. Guardabassi
Affiliation:
Institute di Elettrotecnica ed Elettronica del Politecnico di Milano, Milano, Italy
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The problem of counting partial subgraphs (or patterns, for short) in a given graph has been approached by several mathematicians from various points of view (see, e.g., [1; 3; 5; 13-15; 17-23; 26-29]; applications may also be found in [2; 8; 9; 16]). Specific algorithms have been presented and almost all of them are essentially based upon a careful analysis of the graph under consideration. In these cases, we say that a direct approach has been followed. Unfortunately, when large graphs are considered, all direct counting methods require rather cumbersome computations. For this reason, during the last few years many efforts have been made in finding suitable indirect counting methods. First, Biondi [5] faced the problem of counting cycles in non-oriented graphs by inspection of the complementary graph. More recently, a number of papers [1; 3; 21; 22; 28] have been concerned with counting trees in classes of non-oriented graphs having complementary graphs with special structural properties. However, to the best of our knowledge, no general indirect counting method is available in the literature.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 1 , 01 February 1970 , pp. 22 - 35
Copyright
Copyright © Canadian Mathematical Society 1970

References

1. Bedrosian, S. D., Formulas for the number of trees in a network, IRE Trans. Circuit Theory (Correspondence) 8 (1961), 363–364.Google Scholar
2. Bedrosian, S. D., Application of linear graphs to multi-level maser analysis•, J. Franklin Inst. 274 (1962), 278–283.Google Scholar
3. Bedrosian, S. D., Generating formulas for the number of trees in a graph, J. Franklin Inst. 277 (1964), 313–326.Google Scholar
4. Berge, C., The theory of graphs and its applications (Methuen, London, 1962).Google Scholar
5. Biondi, E., Numerazione delle maglie in una rete qualsiasi, 1st. Lombardo (Accad. Sci. Lett. Rend. A 91 (1957), 912–926.Google Scholar
6. Biondi, E., Divieti, L., and Guardabassi, G., A paths counting method, Relazione Interna LE 66-7, Istituto di Elettrotecnica ed Elettronica del Politecnico di Milano (Milano, 1966).Google Scholar
7. Divieti, L., Un metodo per la determinazione dei sottografi completi di un grafo assegnato, Relazione Interna LE 65-5 Istituto di Elettrotecnica ed Elettronica del Politecnico di Milano (Milano, 1965).Google Scholar
8. Epstein, V. L., On the application of graph theory to the description and analysis of information flows schemes in control systems, Avtomat. i Telemeh 26 (1965), 1403-1410; translated as Automat. Remote Control 26 (1965), 1378–1383.Google Scholar
9. Flament, C., Nombre de cycles complets dans un reseau de communication, Bull. Centre Etudes Rech. Psych. 3 (1959), 105–110.Google Scholar
10. Guardabassi, G., Counting patterns in graphs: a necessary planarity condition, Relazione Interna LE 66-6, Istituto di Elettrotecnica ed Elettronica del Politecnico di Milano (Milano, 1966).Google Scholar
11. Guardabassi, G., On the number of trees in a network (to appear).Google Scholar
12. Guardabassi, G., Counting constrained routes in complete networks: the H-function, unpublished note.Google Scholar
13. Harary, F. and Ross, I. C., The number of complete cycles in a communication network, J. Social Psych. 40 (1954), 329–332.Google Scholar
14. Katz, L., An application of matrix algebra to the study of human relations within organizations, Mimeographed notes, Institute of Statistics, University of North Carolina, 1950.Google Scholar
15. Kel'mans, A. K., The number of trees in a graph. I, Avtomat. i. Telemeh 26 (1965), 2194– 2204; translated as Automat. Remote Control 26 (1965), 2118-2129 (1966).Google Scholar
16. Kendall, M. G., Rank correlation methods (C. Griffin, London, 1948).Google Scholar
17. Kendall, M. G., Further contributions to the theory of paired comparisons, Biometrics 11 (1955), 43–62.Google Scholar
18. Luce, R. D. and Perry, A. D., A method of matrix analysis of group structure, Psychometrika 14 (1949), 95–116.Google Scholar
19. Lunelli, L., Numerazione delle maglie in una rete compléta, 1st. Lombardo Accad. Sci. Lett. Rend. A 91 (1957), 903–911.Google Scholar
20. Nakagawa, N., On evaluation of the graph trees and the driving point admittance, IRE Trans. Circuit Theory 5 (1958), 122–127.Google Scholar
21. O'Neil, P. V., The number of trees in a certain network, Amer. Math. Soc. Meeting, 604, Brooklyn, New York, 1963; Notices Amer. Math. Soc. 10 (1963), 569.Google Scholar
22. O'Neil, P. V. and Slepian, P., The number of trees in a network, IEEE Trans. Circuit Theory 15 (1966), 271–281.Google Scholar
23. O'Neil, P. V. and Slepian, P., An application of Feussner's method to tree counting, IEEE Trans. Circuit Theory (Correspondence) 13 (1966), 336–339.Google Scholar
24. Pototchi, A., A simple algorithm for determining the number of paths in a finite graph, Economic Computation and Economic Cybernetics Studies and Research 2 (1967), 81–85.Google Scholar
25. Riordan, J., An introduction to combinatorial analysis (Wiley, New York, 1958).Google Scholar
26. Ross, I. C. and Harary, F., On the determination of redundancies in sociometric chains, Psychometrika 17 (1952), 195–208.Google Scholar
27. Trent, H. M., A note on the enumeration and listing of all possible trees in a connected linear graph, Proc. Nat. Acad. Sci. 40 (1954), 1004–1007.Google Scholar
28. Weinberg, L., Number of trees in a graph, IRE Proc. (Correspondence) 46 (1958), 1954–1955.Google Scholar
29. Wilf, H. S., A mechanical counting method and combinatorial applications, J. Combinatorial Theory 4 (1968), 246–258.Google Scholar