Degree-similar Unicyclic Graphs are Isomorphic

Two graphs are degree-similar if their adjacency matrices and degree matrices are simultaneously similar.

Godsil and Sun asked whether non-isomorphic degree-similar unicyclic graphs exist.

We prove that they do not exist.

Every graph degree-similar to a unicyclic graph is isomorphic to it.

The proof uses a symbolic leaf-peeling algorithm to recover the rooted trees attached to the unique cycle, and a rigidity lemma for colored cycles to identify automorphisms of the cycle.