Graph invariants are properties associated with a graph that do not change across all possible isomorphisms of the graph. In other words, if a set of graphs are isomorphic, then they all have the same values for certain properties which are called invariants. Examples of graph invariants are the following:

- no of nodes
- no of edges
- the degree distribution (or degree sequence)
- vertex (edge) chromatic number: the minimum no. of colours needed to colour all vertices (edges) such that adjacent vertices (edges) do not have the same colour
- vertex (edge) covering number: the minimum no. of vertices (edges) needed to cover all edges (vertices)

As you can see, the invariants are numbers. A graph invariant *describes* a graph in terms of a simple number. Given a graph, it has a unique number of nodes, number of edges, degree sequence etc.. They find application in chemistry where large chemical compounds are indexed based on these numbers.

It should be noted that the incidence and adjacency matrices are *not* graph invariants. This is because, when isomorphism is computed, if for every pair of adjacent nodes in graph-1, you can find a pair of adjacent nodes in graph-2, and vice versa, the two graphs are isomorphic; labels are not important.

Also, for a given graph, there are unique invariants. However, the converse need not hold. For example, consider the invariant degree sequence. Shown below are two graphs that have the same degree sequence, but are non-isomorphic.

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What I am interested in finding what can be called graph “identifiers”. Graph identifiers are properties or metrics that uniquely identify a graph. Examples of identifiers are adjacency and incidence matrices. Given an adjacency matrix, there is only one graph corresponding to it. Similarly, are there more identifiers? It need not be one property; it can be a combination of a set of properties like the no. of edges, degree sequence, diameter, centrality sequence and so on. What is perhaps also interesting is to measure the “uniqueness” of sets of properties. Uniqueness indicates in percentage terms “the extent” to which a property or a set of properties identify a graph. For instance, the adjacency matrix has a uniqueness value of 1. Again, this can be measured in terms of, say edit distances.

I am not sure there is much work in this direction. Do you think this is an interesting problem?

A consequence of this is graph reconstruction. If we have a set of properties that identify a graph, it is possible to construct an isomorphic graph using those properties. It can be used to recover networks that are subject to disruptions. (My usage of reconstruction is similar to that in literature, but slightly different.)

>> Do you think this is an interesting problem?

Definitely. In fact, when I read the italicised “describes” following “a graph invariant”, I was wondering about things that could “define” a graph.

Yeah. Perhaps we should have a look at spectral graph theory. There they talk about things like eigenvalues and characteristic equations of adjacency matrices.