Relationship between Las Vegas algorithms and deterministic algorithms I'm wondering why the following argument doesn't work for showing that the existence of a Las Vegas algorithm also implies the existence of a deterministic algorithm:

Suppose that there is a Las Vegas algorithm $A$ that solves some graph problem $P$, i.e., $A$ takes an $n$-node input graph $G$ as input (I'm assuming the number of edges is $\le n$) and eventually yields a correct output, while terminating within time $T(G)$ with some nonzero probability.

Suppose that there is no deterministic algorithm that solves $P$. Let $A^\rho$ be the deterministic algorithm that is given by running the Las Vegas algorithm $A$ with a fixed bit string $\rho$ as its random string. 
Let $k=k(n)$ be the number of $n$-node input graphs (with $\le n$ edges).
Since there is no deterministic algorithm for $P$, it follows that, for any $\rho$, the deterministic algorithm $A^\rho$ fails on at least one of the $k$ input graphs. Returning to the Las Vegas algorithm $A$, this means that $A$ has a probability of failure of $\ge 1/k$, a contradiction to $A$ being Las Vegas.
There is no contradiction here. A Las Vegas algorithm terminates in expected polynomial time, say $Q(n)$. Markov's inequality shows that for any given polynomial $R(n)$, it terminates in polynomial time $Q(n)R(n)$ with probability $1-1/R(n)$. The number of graphs on $n$ vertices, $k(n)$, is exponential in $n$, so the failure probability $1/R(n)$ is much larger than $1/k(n)$.

Another subtle point is uniformity. A deterministic algorithm needs to handle any input size, while your $A^\rho$ can only handle inputs of certain size. In general, a randomized algorithm can use an unbounded number of random bits. However, your argument can still be formulated in terms of non-uniform models such as circuits, and it is still wrong for the reason described above.