Agelos Georgakopoulos

Uniqueness of electrical currents in a network of finite total resistance

We show that if the sum of the resistances of an electrical network N is finite, then there is a unique electrical current in N, provided that we do not allow, in a sense made precise in the paper, any flow to escape to infinity.

Hitting Times, Cover Cost, and the Wiener Index of a Tree

We exhibit a close connection between hitting times of the simple random walk on a graph, the Wiener index, and related graph invariants. In the case of trees, we obtain a simple identity relating hitting times to the Wiener index. It is well known that the vertices of any graph can be put in a linear preorder so that vertices appearing earlier in the preorder are “easier to reach” by a random walk, but “more difficult to get out of.” We define various other natural preorders and study their relationships. These preorders coincide when the graph is a tree, but not necessarily otherwise. Our treatise is self-contained, and puts some known results relating the behavior or random walk on a graph to its eigenvalues in a new perspective.

The Planar Cubic Cayley Graphs

We obtain a complete description of the planar cubic Cayley graphs, providing an explicit presentation and embedding for each of them. This turns out to be a rich class, comprising several infinite families. We obtain counterexamples to conjectures of Mohar, Bonnington and Watkins. Our analysis makes the involved graphs accessible to computation, corroborating a conjecture of Droms.

Characterising planar Cayley graphs and Cayley complexes in terms of group presentations

We prove that a Cayley graph can be embedded in the Euclidean plane without accumulation points of vertices if and only if it is the 1-skeleton of a Cayley complex that can be embedded in the plane after removing redundant simplices. We also give a characterisation of these Cayley graphs in term of group presentations, and deduce that they can be effectively enumerated.

Cycle decompositions: From graphs to continua

We generalise a fundamental graph-theoretical fact, stating that every element of the cycle space of a graph is a sum of edge-disjoint cycles, to arbitrary continua. To achieve this we replace graph cycles by topological circles, and replace the cycle space of a graph by a new homology group for continua which is a quotient of the first singular homology group H1. This homology seems to be particularly apt for studying spaces with infinitely generated H1, e.g. infinite graphs or fractals.

Lamplighter graphs do not admit harmonic functions of finite energy

We prove that a lamplighter graph of a locally finite graph over a finite graph does not admit a non-constant harmonic function of finite Dirichlet energy.

Every rayless graph has an unfriendly partition

We prove that every rayless graph has an unfriendly partition.

The planar cubic Cayley graphs of connectivity 2

Abstract We classify the planar cubic Cayley graphs of connectivity 2, providing an explicit presentation and embedding for each of them. Combined with Georgakopoulos (2017) this yields a complete description of all planar cubic Cayley graphs.

New Bounds for Edge-Cover by Random Walk

We show that the expected time for a random walk on a (multi-)graph

On covers of graphs by Cayley graphs

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