Consider a finite connected graph.
By Cauchy-Schwarz and the handshake lemma, it is easy to see that $\left( \sum_{i=1}^n \sqrt{d_i} \right)^2 \leq n \sum_{i=1}^n d_i =2mn$, with equality iff the graph is regular (constant degree).
Here $d_i$ is the degree of vertex $i$, $m$ is the number of edges and $n$ is the number of vertices.
Is there a good lower bound for a simple connected graph? I wonder if the lower bound might be half the upper bound?
Best Answer
Given a fixed number of edges and vertices, Linial and Rozenman conjectured that the graphs which minimize $\sum_i \sqrt{d_i}$ are the ones given by taking the largest possible complete graph together with an extra vertex of remaining degree and the rest as isolated vertices. Their conjecture was proved in "Minimizer graphs for a class of extremal problems" by Dan Ismailescu and Dan Stefanica, Journal of Graph Theory (39), 230–240, 2002.
Since you are interested in connected graphs, perhaps that is of little use to you. I will mention one more paper which gives a lower bound on $\sum_i \sqrt{d_i}$. See "Sums of powers of the degrees of a graph" by S. M. Cioabă (Theorem 8), which gives a lower bound in terms of the number of vertices, edges, and minimum and maximum degree.
Finally there is also the amusing lower bound by Székely, Clark and Entriger $$\left(\sum_{i=1}^n \sqrt{d_i}\right)^2\geq \sum_{i=1}^n d_i^2.$$