The quasi-Wiener and the Kirchhoff indices coincide

Journal of Chemical Information and Computer Sciences

Volumn 36, Issue 5, 1996, Pages 982-985

  Gutman, Ivan ^a,c   Mohar, Bojan ^b  

^a UNIVERSITY OF SZEGED   (Hungary)

^b UNIVERSITY OF LJUBLJANA   (Slovenia)

^c UNIVERSITY OF KRAGUJEVAC   (Serbia)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0010401244     PISSN: 00952338     EISSN: None     Source Type: Journal    
DOI: 10.1021/ci960007t     Document Type: Article

References (23)

1. While refereeing this paper Douglas J. Klein (Texas A&M University, Galveston) pointed out that he has noted the identity W* = Kf to a few people over the last two years and that this identity is so noted in a paper by Zhu et al. "Extensions of the Wiener Number" which, after this paper had been submitted and accepted for publication, appeared in J. Chem. Inf. Comput. Sci. 1996, 36, 420-428.
2. Grone, R.; Merris, R.; Sunder, V. S. The Laplacian Spectrum of a Graph. SIAM J. Matrix Anal. Appl. 1990, 11, 218-238.
3. Mohar, B. The Laplacian Spectrum of Graphs. In Graph Theory, Combinatorics, and Applications; Alavi, Y., Chartrand, G., Ollermann, O. R., Schwenk, A. J., Eds.; Wiley: New York, 1991; pp 871-898.
4. Mohar, B.; Poljak, S. Eigenvalues in Combinatorial Optimization. In Combinatorial and Graph-Theoretical Problems in Linear Algebra; Brualdi, R. A., Friedland, S., Klee, V., Eds.; Springer-Verlag: Berlin, 1993; pp 107-151.
5. Merris, R. Laplacian Matrices of Graphs: A Survey. Linear Algebra Appl. 1994, 197-198, 143-176.
6. Merris, R. The Distance Spectrum of a Tree. J. Graph Theory 1990, 14, 365-369.
7. In ref 3. B. McKay's private communication is given as the source of formula 1.
8. Mohar, B. Eigenvalues. Diameter, and Mean Distance in Graphs. Graphs Combin. 1991, 7, 53-64.
9. Merris, R. An Edge Version of the Matrix-Tree Theorem and the Wiener Index. Lin. Multilin. Algebra 1989, 25, 291-296.
10. Mohar, B.; Babić, D.; Trinajstić, N. A Novel Definition of the Wiener Index for Trees. J. Chem. Inf. Comput. Sci. 1993, 33, 153-154.
11. Gutman, I.; Yeh, Y. N.; Lee, S. L.; Luo, Y. L. Some Recent Results in the Theory of the Wiener Number. Indian J. Chem. 1993, 32A, 651-661.
12. Gutman, I.; Lee, S. L.; Chu, C. H.; Luo, Y. L. Chemical Applications of the Laplacian Spectrum of Molecular Graphs: Studies of the Wiener Number. Indian J. Chem. 1994, 33A, 603-608.
13. Marković, S.; Gutman, I.; Bancević, Z. Correlation between Wiener and Quasi-Wiener Indices in Benzenoid Hydrocarbons. J. Serb. Chem. Soc. 1995, 60, 633-636.
14. Klein, D. J.; Randić, M. Resistance Distance. J. Math. Chem. 1993, 12, 81-95.
15. Bonchev, D.; Balaban, A. T.; Liu, X.; Klein, D. J. Molecular Cyclicity and Centricity of Polycyclic Graphs. I. Cyclicity Based on Resistance Distances or Reciprocal Distances. Internat. J. Quantum Chem. 1994, 50, 1-20.
16. In ref 15 the Kirchhoff index is defined as twice the sum of resistance distances between all pairs of vertices of a graph, being thus inconsistent with the usual definition of the Wiener index. In order to maintain a full analogy with W. in this work we define the Kirchhoff index Kf as just the sum of resistance distances between all pairs of vertices. Then, in particular, W = Kf holds for trees.
17. Seshu, S.; Reed, M. B. Linear Graphs and Electrical Networks; Addison-Wesley: Reading, 1961.
18. Edminister, J. A. Electric Circuits; McGraw-Hill: New York, 1965.
19. Ben-Israel, A.; Greville, T. N. E. Generalized Inverses - Theory and Applications; Wiley: New York, 1974.
20. Campbell, S. L.; Meyer, C. D. Generalized Inverses of Linear Transformations; Pitman: London, 1979.
21. † for the generalized inverse is taken from ref 19; it should not be confused with the Hermitean conjugate, a notation often used in theoretical chemistry and theoretical physics.
22. The generalized inverse considered in this paper was first invented by R. H. Moore in 1935 but was eventually more or less forgotten. An equivalent concept was introduced by R. Penrose in 1955, who was apparently unaware of Moore's work. For additional details see pp 9-11 of ref 20.
23. T·e = 1.