= Solution
For a $\Delta$-regular undirected <graph>, its <Graph Laplacian> is $L=\Delta I-A$, and for any complex vector $v$,
$$
v^*Lv=\sum_{\{i,j\}\in E}|v_i-v_j|^2.
$$
If $Av=\Delta v$, the left side is zero, hence $v_i=v_j$ on every <edge>. In a <connected graph> all coordinates coincide. Thus the <eigenspace> is exactly the constant vectors, of dimension one. Since the <adjacency matrix of a graph> is symmetric and diagonalizable,
$$
\boxed{\text{the algebraic and geometric multiplicities of }\Delta\text{ are one}.}
$$
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