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

If $Av=\lambda v$, choose $i$ with $|v_i|$ maximal. Then $|\lambda ||v_i|=|{\sum }_{j\sim i}{v}_j|\le d(i)|v_i|\le \Delta |v_i|$, so $|\lambda |\le \Delta$.

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.