Abstract
The walk distances in graphs have no direct interpretation in terms of walk weights, since they are introduced via the logarithmsof walk weights. Only in the limiting cases where the logarithms vanish such representations follow straightforwardly. The interpretation proposed in this chapter rests on the identity \(\ln \det B = tr \ln B\)applied to the cofactors of the matrix \(I - tA,\)where Ais the weighted adjacency matrix of a weighted multigraph and tis a sufficiently small positive parameter. In addition, this interpretation is based on the power series expansion of the logarithm of a matrix. Kasteleyn (Graph theory and crystal physics. In: Harary, F. (ed.) Graph Theory and Theoretical Physics. Academic Press, London, 1967) was probably the first to apply the foregoing approach to expanding the determinant of I− A. We show that using a certain linear transformation the same approach can be extended to the cofactors of I− tA, which provides a topological interpretation of the walk distances.
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Notes
- 1.
In this chapter, a distanceis assumed to satisfy the axioms of metric.
- 2.
In the more general case of weighted digraphs, the ij-entry of the matrix \({R}_{t} - I\)is called the Katz similaritybetween vertices iand j. Katz [14] proposed it to evaluate the social status taking into account all \(i \rightarrow j\)paths. Among many other papers, this index was studied in [13, 23].
- 3.
Cf. “dihedral equivalence” in [11].
- 4.
A cyclic sequenceis a set \(X =\{ {x}_{1},\ldots ,{x}_{N}\}\)with the relation “next” \(\eta =\{ ({x}_{2},{x}_{1}),\ldots , ({x}_{N},{x}_{N-1}),\) \(({x}_{1},{x}_{N})\}\).
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Chebotarev, P., Deza, M. (2013). A Topological Interpretation of the Walk Distances. In: Mucherino, A., Lavor, C., Liberti, L., Maculan, N. (eds) Distance Geometry. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-5128-0_7
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