Harmonic morphisms of graphs. Part I: Graph coverings (Q2810314)

Let \(G\), \(G^\prime\) be graphs. A function \(\varphi : V(G)\cup E(G) \rightarrow V(G^\prime) \cup E(G^\prime\) is said to be a morphism from \(G\) to \(G^\prime\) if \(\varphi(V(G)) \in V(G^\prime)\), and for every edge \(e \in E(G)\) with endpoints \(x\) and \(y\), either \(\varphi(e) \in E(G^\prime)\) and \(\varphi(x)\), \(\varphi(y)\) are the endpoints of \(\varphi(e)\), or \(\varphi(e) \in V(G^\prime)\) and \(\varphi(e) = \varphi(x) = \varphi(y)\). We write \(\varphi : G \rightarrow G^\prime\) for brevity. If \(\varphi(E(G)) \subseteq E(G^\prime) \) then we say that \(\varphi\) is a homomorphism. A bijective homomorphism is called an isomorphism, and an isomorphism \(\varphi: G \rightarrow G\) is called an automorphism. A morphism \(\varphi : G \rightarrow G^\prime\) is said to be harmonic if, for all \(x \in V(G), y \in V(G^\prime)\) such that \(y = \varphi(x)\), the quantity \(|e \in E(G)\mid x \in e, \varphi(e) \in e^\prime|\) is the same for all edges \(e^\prime \in E(G^\prime)\) such that \(y \in e^\prime\). Harmonic morphisms of graphs are a natural discrete analogue of homomorphic maps between Riemann surfaces. Harmonic maps are generalisation of graph coverings, the simplest examples of harmonic maps are any covering of graphs, and a natural projection of the wheel graph \(W_6\) onto the wheel graph \(W_2\).NEWLINENEWLINEThis book is the first part of the book ``Harmonic Morphisms''. The authors splits this volume into three chapters include ``graphs and groups'', ``graph coverings and the voltage spaces'' and ``applications of graph covering''; this book is basic for the study in the second part: the lifting automorphism and the generalized graph coverings to harmonic morphisms.