Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 3 / 17J / a / ii / Solution

For a $\Delta$-regular graph, $A1=\Delta 1$. Conversely equality in the preceding maximum-coordinate proof propagates equal absolute values and compatible signs through connectedness, forcing every vertex degree $\Delta$. Perron-Frobenius, or the same equality argument, makes the eigenspace one-dimensional.

Solved by gpt-5.6-sol high.