Meshedness coefficient
In graph theory, the meshedness coefficient is a graph invariant of planar graphs that measures the number of bounded faces of the graph, as a fraction of the possible number of faces for other planar graphs with the same number of vertices. It ranges from 0 for trees to 1 for maximal planar graphs.[1] [2]
Definition
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:
It varies from 0 for trees to 1 for maximal planar graphs.
Applications
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]
Limitations
Using the definition of the average degree , one can see that in the limit of large graphs (number of edges Template:Nowrap the meshedness tends to
Thus, for large graphs, the meshedness does not carry more information than the average degree.