If $G_1$ and $G_2$ are two regular graphs on same no. of vertices,with same regularity and have identical Laplacian spectrum, are $G_1$ and $G_2$ isomorphic?
2026-03-25 16:00:00.1774454400
Are $G_1$ and $G_2$ isomorphic?
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Not necessarily. Notice that $k$-regular graphs on the same number of vertices have the same Laplacian spectrum if and only if they have the same spectrum on their adjacency matrices. In particular, $\lambda$ is an eigenvalue of the adjacency matrix iff $k-\lambda$ is an eigenvalue of the Laplacian matrix.
On the other hand, strongly-regular graphs form a rich class of graphs containing many cospectral, non-isomorphic $k$-regular graphs on $n$ vertices, for some values of $n$ and $k$. The smallest example in this class occurs when $n=16$ and $k=6$, and their adjacency matrices can be found at Ted Spence's site here.