I have an undirected weighted graph and I need to formally describe it. Abstraction via automata or labelled transition systems seems to be not defined for undirected graphs, only directed graphs are covered. States within the graph are dependent on one another, however direction itself is not relevant.
Do you have any idea what mathematical model could be used to formally describe such graph?
You could consider an undirected weighted graph to be a directed weighted graph with the restriction that for any nodes A and B, if there exists an edge from A to B then there also exists an edge with the same weight from B to A. So any model for a directed graph can also be used for an undirected graph. So if you use an matrix that shows the weights, just require that the matrix be symmetric. If you use a function from ordered pairs to real numbers, instead use a function from unordered pairs (here I mean a multiset of size two) or sets of size one or two. And so on.
For a programming model (this is a programming site, after all) you could use a symmetric matrix of weights. In some languages like Python where you can define a list of lists with varying lengths, you could use a triangular matrix rather than a symmetric one, showing an edge only from a node listed earlier to a node listed later. This might be easier than enforcing the symmetric restriction on all matrix changes.
Graphs are a mathematical abstraction. Directed labelled graphs are not modelled as automata, but the other way round.
If you are looking for an “implementation” within “lower-level mathematics”, there are various possibilities for that:
Every undirected graph can be seen as a directed graph subject to the condition that whenever there is an edge from a to b, there is an edge from b to a. In the case of weighted graphs, we add the requirement that antiparallel edges have the same weight.
Ever since ZFC (and possibly before that) mathematicians have a tendency to model stuff as sets. Then, a simple undirected weighted Graph would be a triple (V, E, w) with V the set of your vertices / nodes, E a subset of the powerset of V containing only two-elemental sets (The set {a, b} represents the edge between a and b.) and a weight function w: E -> IR.