For instance, in the path graph $P_4$, node $1$ and $4$ are symmetric, how to mathematically describe this in graph theory? And any algebraic properties related to this? Thanks!
2026-04-23 06:31:04.1776925864
How to describe symmetric nodes in a graph
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I think the property you are looking for is an automorphism. An isomorphism is a homomorphism on a graph that is a bijection. It is the generally accepted notion of graph equivalence. On graphs, homomorphisms are functions that preserve adjacencies. An automorphism is an isomorphism of the graph to itself.
Are you familiar with symmetry groups? The idea of a symmetry group on a graph is to permute the vertices. A permutation that gives an isomorphism is called an automorphism. So Aut(G), the automorphism group of the graph, is a subgroup of Sym(G), the symmetry group on the graph.
Note as well this is more of an abstract algebra topic than a linear algebra topic. I noticed you tagged this with "matrices", so I just wanted to clarify.