We present a simple nonadaptive randomized algorithm that estimates the number of edges in a simple, unweighted, undirected graph, possibly containing isolated vertices, using only degree and random edge queries. For an n-vertex graph, our method requires only O(√(n)) queries, achieving sublinear query complexity. The algorithm independently samples a set of vertices and queries their degrees, and also independently samples a set of edges, using the answers to these queries to estimate the total number of edges in the graph. We further prove a matching lower bound, establishing the optimality of our algorithm and resolving the non-adaptive query complexity of this problem with respect to degree and random-edge queries.
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