Norinvariant

Norinvariant is a graph-theoretic invariant used to characterize finite graphs in combinatorics and network analysis. It is defined for an undirected simple graph G = (V, E) by associating to each edge a local contribution that depends on the degrees of its endpoints. The Norinvariant N(G) is obtained by summing these local contributions over all edges: N(G) = sum over {u, v} in E of φ(deg(u), deg(v)), where φ is a symmetric function chosen to reflect particular structural properties of interest. Because the definition relies only on vertex degrees and edge endpoints, N(G) is invariant under graph isomorphism and labeling.

Choice of φ determines the sensitivity of the Norinvariant to degree structure. Common choices include φ(a, b) = a + b, φ(a, b) = min(a, b), and φ(a, b) = |a − b|, each highlighting different aspects of degree correlations in the graph. A regular graph with degree k yields N(G) = |E|·φ(k, k), illustrating how the invariant interacts with global regularity.

Properties and limitations: The Norinvariant is additive over disjoint unions (N(G1 ∪ G2) = N(G1) + N(G2)) and is robust under relabeling. It is not a complete invariant; distinct non-isomorphic graphs can share the same Norinvariant for a given φ. Its discriminatory power thus depends on the chosen local function and the graph class under study.

Applications: It has been proposed for distinguishing graphs with identical degree sequences but different degree assortativity, summarizing network structure in chemical graph theory, and serving as a feature in graph classification tasks where degree-based information is informative.

See also: Graph invariant, Degree sequence, Graph isomorphism, Degree assortativity.