If you take a complete graph on 11 nodes, it will have 55 edges, and every node will have degree 10. To construct a graph with 53 edges we must remove two edges. There are two cases to consider: either the two edges share a node or they don't.
Case 1: Call three of the nodes $A$, $B$, and $C$. Remove edges $AB$ and $BC$. Now $A$ and $C$ have degree 9, $B$ has degree 8 and all other nodes have degree 10. The graph remains connected, so there is an Eulerian path from $A$ to $C$ but there is no Eulerian cycle.
Case 2: Remove two disjoint edges $AB$ and $CD$ (where $D$ is a fourth node) Now there are four nodes of degree 9 so there is not even an Eulerian path.
In both cases there is no Eulerian cycle.