Filtering graphs to check isomorphism and extracting mapping by using the Conductance Electrical Model

• The method uses the CEM to model the graphs and detect the isomorphic graphs.
• The time complexity for detecting non-isomorphic graphs of N degree is in most of the cases O(N3).The CEM can obtain the complete or partial node mapping between the graphs.
• The time complexity for detecting graphs isomorphism in the worst time complexity can be exponential with respect to the number of repeated star values.The detection and mapping of a graph isomorphism (excluding co-resistances) are clearly separated, which allows doing the filter process in two phases.
• The filter performs an early detection of non-isomorphic graphs.
• The filtering process is not probabilistic, not iterative neither recursive, so the computation complexity is deterministic and only depends on N.
• We do not need to calculate the pseudoinverse, we can compute the star resistance values by a sum of finite number of terms.

This paper presents a new method of filtering graphs to check exact graph isomorphism and extracting their mapping. Each graph is modeled by a resistive electrical circuit using the Conductance Electrical Model (CEM). By using this model, a necessary condition to check the isomorphism of two graphs is that their equivalent resistances have the same values, but this is not enough, and we have to look for their mapping to find the sufficient condition. We can compute the isomorphism between two graphs in O(N3)O(N3), where N   is the order of the graph, if their star resistance values are different, otherwise the computational time is exponential, but only with respect to the number of repeated star resistance values, which usually is very small. We can use this technique to filter graphs that are not isomorphic and in case that they are, we can obtain their node mapping. A distinguishing feature over other methods is that, even if there exists repeated star resistance values, we can extract a partial node mapping (of all the nodes except the repeated ones and their neighbors) in O(N3)O(N3). The paper presents the method and its application to detect isomorphic graphs in two well know graph databases, where some graphs have more than 600 nodes.