The meshedness coefficient is used to compare the general cycle structure of a connected planar graph to two extreme relevant references. In one end, there are trees, planar graphs with no cycle.^[1] The other extreme is represented by maximal planar graphs, planar graphs with the highest possible number of edges and faces for a given number of vertices. The normalized meshedness coefficient is the ratio of available face cycles to the maximum possible number of face cycles in the graph. This ratio is 0 for a tree and 1 for any maximal planar graph.

More generally, it can be shown using the Euler characteristic that all n-vertex planar graphs have at most 2n − 5 bounded faces (not counting the one unbounded face) and that if there are m edges then the number of bounded faces is m − n + 1 (the same as the circuit rank of the graph). Therefore, a normalized meshedness coefficient can be defined as the ratio of these two numbers:

${\displaystyle \alpha ={\frac {m-n+1}{2n-5}}.}$ 

It varies from 0 for trees to 1 for maximal planar graphs.

The meshedness coefficient can be used to estimate the redundancy of a network. This parameter along with the algebraic connectivity which measures the robustness of the network, may be used to quantify the topological aspect of network resilience in water distribution networks.^[3] It has also been used to characterize the network structure of streets in urban areas.^[4][5][6]

Using the definition of the average degree ${\displaystyle \langle k\rangle =2m/n}$ , one can see that in the limit of large graphs (number of edges ${\displaystyle n\gg 1}$ ) the meshedness tends to

${\displaystyle \alpha \approx {\frac {\langle k\rangle }{4}}-{\frac {1}{2}}}$ 

Thus, for large graphs, the meshedness does not carry more information than the average degree.