In the Zeta functions of graphs : a stroll through the garden 's 99 page , there is the define of conjugacy class, but I can't understand it well. Are there any difference between conjugacy class and homotopy class?
2025-01-13 02:25:07.1736735107
What are the conjugacy class and homotopy class of fundamental group in graph theory?
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A homotopy class is an equivalence class of walks, it's a topological notion. A conjugacy class is a group theoretic notion - the class of $g$ consists of all elements $x^{-1}gx$ for $x$ in the group. In a graph, each homotopy class contains a unique reduced walk, which determines the class. A reduced closed walk is an element of the fundamental group.