degreeL

degreeL is a generalized degree function used in graph theory to quantify the connectivity of a vertex in a labeled graph. The L denotes a set of labels or weights assigned to edges. For a simple undirected graph G=(V,E) with a label function l:E→L and a weight function w:L→R^+, the degreeL of a vertex v is defined as degL(v) = sum over all edges e incident to v of w(l(e)). In the common case where all edges carry the same label or where w is the identity, degL becomes the standard degree. In directed graphs, one can define in-degreeL and out-degreeL analogously, or define a total degreeL by summing both.

Variants include normalization by the number of incident edges, or by the maximum possible degree in a given graph; degreeL can also be defined with edge multiplicities or by using a more complex function of labels, such as degL(v) = sum w(l(e)) where w is nonlinear.

Applications include network analysis and ranking, where degreeL provides a more nuanced measure of a node’s influence by incorporating edge labels such as interaction strength, capacity, or trust scores. It is often used in conjunction with standard degree and other centrality measures to study labeled or weighted networks.

The concept is not universal; degreeL appears in different forms in research on labeled graphs and weighted networks, where the choice of label set and weight function can reflect domain-specific semantics. See also: degree, weighted degree, centrality, graph labeling.