According to answers here https://math.stackexchange.com/questions/1524598/a-general-incidence-problem// an unigraph comes from unigraphic degree sequences if it can be uniquely determined by its degree sequences.

Adjacency matrix or incidence matrix of a graph provides every computable information about a graph.

Apart from degree sequences what additional information would one need to provide to capture any graph uniquely? That is degree sequence plus some additional information should give as much information on a graph as possible. What is this additional information?

Adjacency matrix needs about $2 n^2\log n$ bits of information (to specify a row and column you need to spend $\log n + \log n$ bits and we have $n^2$ combinations of rows and columns). If we choose to represent a graph by fixing an order on vertices, you still need $n^2$ bits (same as adjacency matrix).

Degree sequences need about $n\log n$ bits of information.

So we are missing a factor of $2n$ or $\frac n{\log n}$ depending on how we look.

It is clear degree sequences are insufficient. Information theoretically we need to provide incidence relations. Is there a way to specify this information implicitly so that a multiplicative factor of $n$ is taken care of while we do not add additionally the degree information?

That is can we separate the degree information from this additional *multiplicative* piece? Is there a meaning to this *multiplicative* factor? Thinking of indices and conditional entropies only gives you additive meaning. This means you get only quadratic factors back.