Abstract
It has long been realized that connected graphs have some sort of geometric structure, in that there is a natural distance function (or metric), namely, the shortest-path distance function. In fact, there are several other natural yet intrinsic distance functions, including: the resistance distance, correspondent “square-rooted” distance functions, and a so‐called “quasi‐Euclidean” distance function. Some of these distance functions are introduced here, and some are noted not only to satisfy the usual triangle inequality but also other relations such as the “tetrahedron inequality”. Granted some (intrinsic) distance function, there are different consequent graph-invariants. Here attention is directed to a sequence of graph invariants which may be interpreted as: the sum of a power of the distances between pairs of vertices of G, the sum of a power of the “areas” between triples of vertices of G, the sum of a power of the “volumes” between quartets of vertices of G, etc. The Cayley–Menger formula for n-volumes in Euclidean space is taken as the defining relation for so-called “n-volumina” in terms of graph distances, and several theorems are here established for the volumina-sum invariants (when the mentioned power is 2).
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References
A.T. Balaban, Topological indices based on topological distances in molecular graphs, Pure Appl. Chem. 55 (1983) 199–206.
L.M. Blumenthal, New theorems and methods in determinant theory, Duke Math. J. 2 (1936) 396–404.
L.M. Blumenthal, Distance Geometry (Chelsea, New York, 1970).
F. Buckley and F. Harary, Distance in Graphs (Addison-Wesley, Reading, MA, 1989).
A. Cayley, A theorem on the geometry of position, Cambridge Math. J. 2 (1841) 267–271.
K.L. Collins, Factoring distance matrix polynomials, Discrete Math. 122 (1993) 103–112.
J.K. Doyle and J.E. Graver, Mean distance in a graph, Discrete Math. 17 (1977) 147–154.
P.G. Doyle and J.L. Snell, Random Walks and Electrical Networks (Math. Assoc. Amer., Washington, DC, 1984).
R.C. Entringer and D.E. Jackson, Distance in graphs, Czechoslovak Math. J. 26 (1976) 283–296.
M. Fiedler, Algebraic connectivity of graphs, Czechoslovak Math. J. 23 (1973) 298–305.
M. Fiedler, A geometric approach to the Laplacian matrix of a graph, in: Combinatorial and Graph-Theoretical Problems in Linear Algebra, eds. R.A. Brualdi, S. Friedland and V. Klee (Springer, Berlin, 1993) pp. 73–98.
R.L. Graham, A.J. Hoffman and H. Hosoya, On the distance matrix of a directed graph, J. Graph Theory 1 (1977) 85–88.
R.L. Graham and L. Lovasz, Distance matrix polynomials of trees, Adv. Math. 29 (1978) 60–88.
I. Gutman, Y.-N. Yeh, S.-L. Lee and Y.-L. Luo, Some recent results in the theory of the Wiener number, Indian J. Chem. 32A (1993) 651–661.
H. Hosoya, On some counting polynomials in chemistry, Discrete Appl. Math. 19 (1988) 239–257.
D.J. Klein, Geometry, graph metrics, and Weiner, Comm. Math. Chem. 35 (1997) 7–27.
D.J. Klein, I. Lukovits and I. Gutman, On the definition of the hyper-Wiener index for cycle-containing tructures, J. Chem. Inf. Comput. Sci. 35 (1995) 50–52.
D.J. Klein and M. Randić, Resistance distance, J. Math. Chem. 12 (1993) 81–95.
I. Lukovits, A note on a formula for the hyper-Wiener index of some trees, J. Chem. Inf. Comput. Sci. 34 (1994) 1079–1081.
K. Menger, Untersuchungen uber allgemeine Metrik, Math. Ann. 100 (1928) 75–163.
K. Menger, New foundation of Euclidean geometry, Amer. J. Math. 53 (1931) 721–745.
R. Merris, An edge version of the matrix-tree theorem and the Wiener index, Linear and Multilinear Algebra 25 (1989) 291–296.
B. Mohar, Eigenvalues, diameter, and mean distance in graphs, Graphs Combin. 7 (1991) 53–64.
D.S. Mitoinovic, J.E. Pecaric and V. Volenec, Recent Advances in Geometric Inequalities (Kluwer, Dordrecht, 1989) p. 551.
J. Plesnik, On the sum of all distances in a graph or digraph, J. Graph Theory 8 (1984) 1–21.
O.E. Polansky and D. Bonchev, The Wiener number of graphs. I, Comm. Math. Chem. 21 (1986) 133–186.
M. Randić, Novel molecular description for structure-property studies, Chem. Phys. Lett. 211 (1993) 478–483.
M. Randić, Molecular shape profiles, J. Chem. Inf. Comput. Sci. 35 (1995) 373–382.
D.H. Rouvray, The role of the topological distance matrix in chemistry, in: Mathematical and Computational Concepts in Chemistry, ed. N. Trinajstić (Ellis Horwood, Chichester, 1986) pp. 295– 306.
L.W. Shapiro, An electrical lemma, Math. Mag. 60 (1987) 36–38.
N. Trinajstić, Chemical Graph Theory (CRC Press, Boca Raton, FL, 1983).
H. Wiener, Structural determination of paraffin boiling points, J. Am. Chem. Soc. 69 (1947) 17–20.
H. Wiener, Correlation of heats of isomerization and differences in heats of vaporization of isomers, J. Am. Chem. Soc. 69 (1947) 2636–2638.
W.A. Wilson, A relation between metric and Euclidean spaces, Amer. J. Math. 54 (1932) 505–517.
H.-Y. Zhu and D.J. Klein, Graph-geometric invariants for molecular structures, J. Chem. Inf. Comput. Sci. 36 (1996) 1067–1075.
H.-Y. Zhu, D.J. Klein and I. Lukovits, Extensions of the Wiener number, J. Chem. Inf. Comput. Sci. 36 (1996) 420–428.
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Klein, D., Zhu, H. Distances and volumina for graphs. Journal of Mathematical Chemistry 23, 179–195 (1998). https://doi.org/10.1023/A:1019108905697
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DOI: https://doi.org/10.1023/A:1019108905697