Iswgraphs

Iswgraphs are a class of finite, undirected graphs equipped with a weight function on edges and a vertex indexing. Formally, an Iswgraph is a quadruple (V,E,w,idx) where (V,E) is a simple graph, w:E→R assigns a real weight to each edge, and idx:V→N is a bijection that assigns a unique index to every vertex. The index is used to define canonical local invariants and to specify a notion of isomorphism that preserves both topology and labeling.

The primary concept in Iswgraphs is the isw-sequence. For a vertex v, the isw-sequence S(v) is the multiset of weights of edges incident to v, ordered by the indices of the neighboring vertices. These sequences summarize local structure and are compared under index-preserving isomorphisms. Two Iswgraphs are considered isomorphic if there exists a bijection φ:V→V with idx(φ(v))=idx(v) for all v, such that {u,v}∈E iff {φ(u),φ(v)}∈E and w({u,v})=w({φ(u),φ(v)}) for all edges. The index constraint means that isomorphisms must respect the vertex labeling while preserving edge weights.

Properties of Iswgraphs include the relationship between weight distribution and graph symmetries. When edge weights are all distinct, the structure tends toward rigidity, reducing nontrivial automorphisms. If weights repeat, symmetries can arise even in otherwise asymmetric topologies. Iswgraphs generalize traditional weighted graphs and vertex-labeled graphs by combining edge weights with a unified vertex indexing.

Applications include graph isomorphism testing, pattern recognition, and mathematical studies of invariants in weighted structures. See also weighted graphs, graph isomorphism problem, and vertex labeling.