This paper encompasses a motley collection of ideas from several areas of mathematics, including, in no particular order, random walks, the Picard group, exchange rate networks, chip-firing games, cohomology, and the conductance of an electrical network. The linking threads are the discrete Laplacian on a graph and the solution of the associated Dirichlet problem. Thirty years ago, this subject was dismissed by many as a trivial specialisation of cohomology theory, but it has now been shown to have hidden depths. Plumbing these depths leads to new theoretical advances, many of which throw light on the diverse applications of the theory. 1991 Mathematics Subject Classification 05C50.

Algebraic Potential Theory on Graphs

We describe simple greedy algorithms to construct the shortest set of loops that generates either the fundamental group (with a given basepoint) or the first homology group (over any fixed coefficient field) of any oriented 2-manifold. In particular, we show that the shortest set of loops that generate the fundamental group of any oriented combinatorial 2-manifold, with any given basepoint, can be constructed in O(n log n) time using a straightforward application of Dijkstra's shortest path algorithm. This solves an open problem of Colin de Verdiere and Lazarus.

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Greedy optimal homotopy and homology generators

We study harmonic morphisms of graphs as a natural discrete analogue of holomorphic maps between Riemann surfaces. We formulate a graph-theoretic analogue of the classical Riemann-Hurwitz formula, study the functorial maps on Jacobians and harmonic 1-forms induced by a harmonic morphism, and present a discrete analogue of the canonical map from a Riemann surface to projective space. We also discuss several equivalent formulations of the notion of a hyperelliptic graph, all motivated by the classical theory of Riemann surfaces. As an application of our results, we show that for a 2-edge-connected graph G which is not a cycle, there is at most one involution $\iota$ on G for which the quotient $G/\iota$ is a tree. We also show that the number of spanning trees in a graph G is even if and only if G admits a non-constant harmonic morphism to the graph B_2 consisting of 2 vertices connected by 2 edges. Finally, we use the Riemann-Hurwitz formula and our results on hyperelliptic graphs to classify all hyperelliptic graphs having no Weierstrass points.

Harmonic morphisms and hyperelliptic graphs

We characterize the distributions of short cycles in a large metabolic network previously shown to have small world characteristics and a power law degree distribution. Compared with three classes of random networks, including Erdős–Renyi random graphs and synthetic small world networks of the same connectivity, both the metabolic network and models for the chemical reaction networks of planetary atmospheres have a particularly large number of triangles and a deficit in large cycles. Short cycles reduce the length of detours when a connection is clipped, so we propose that long cycles in metabolism may have been selected against in order to shorten transition times and reduce the likelihood of oscillations in response to external perturbations.

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Relevant cycles in chemical reaction networks

2-connected outerplanar graphs have a unique minimal cycle basis with length $2\vert E\vert-\vert V\vert$. They are the only Hamiltonian graphs with a cycle basis of this length.

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Minimal Cycle Bases of Outerplanar Graphs

Four topological entities are necessary for a complete description of a network graph, as required by orthogonal network theory: the seg, the circ and two others, introduced in this paper, as complementary to the seg and the circ respectively. A vector space and its dual are associated with a nonoriented graph. Both spaces are decomposed in two complementary subspaces. The vectors of each subspace share the topological properties of one of the four entities singled out. The results obtained are extended to a directed graph by associating two dual Z-modules with it. The given description of the network graph is perfectly functional to the orthogonal network theory.

A. Terrusi

Papers

On vector spaces associated with a graph