Best parameterized algorithm for maximum clique
Maximum clique in graphs with degree $d$ can be reduced to $n$ instances of maximum clique in a graph with at most $d$ vertices: for each vertex, compute maximum clique in the induced subgraph of the neighborhood of the vertex.

Therefore, if we omit polynomial factors, the time complexity of maximum clique in graphs with degree at most $d$ is the same as time complexity of maximum clique in graphs with $d$ vertices. I think currently the best known (and peer reviewed) algorithm works in $O(1.1996^n)$ time [1]. Confusingly, there also seems to be an older claimed result with $O(1.1892^n)$ time [2].

[1] M. Xiao and H. Nagamochi. Exact algorithms for maximum independent set.
Information and Computation, 255:126–146, 2017.

[2] J.M Robson. Finding a maximum independent set in time $O(2^{n/4})$. LaBRI, Université de Bordeaux I, Technical report, 2001.