We investigate how the graph topology influences the robustness to noise in undirected linear consensus networks. We measure the structural robustness by using the smallest possible value of steady state population variance of states under the noisy consensus dynamics with edge weights from the unit interval. We derive tight upper and lower bounds on the structural robustness of networks based on the average distance between nodes and the average node degree. Using the proposed bounds, we characterize the networks with different types of robustness scaling under increasing size. Furthermore, we present a fundamental trade-off between the structural robustness and the average degree of networks. While this trade-off implies that a desired level of structural robustness can only be achieved by graphs with a sufficiently large average degree, we also show that there exist dense graphs with poor structural robustness. We then show that, random k-regular graphs (the degree of each node is k) with n nodes typically have near-optimal structural robustness among all the graphs with size n and average degree k for sufficiently large n and k. We also show that when k increases properly with n, random k-regular graphs maintain a structural robustness within a constant factor of the complete graph's while also having the minimum average degree required for such robustness.