Counting loopy graphs with given degrees

TL;DR

This paper derives asymptotically precise formulas for counting symmetric 0-1 matrices with specified row sums, including loops, in both sparse and dense regimes, extending previous zero-diagonal enumeration results.

Contribution

It introduces formulas for counting loopy graphs with given degrees, allowing diagonal entries and covering both sparse and dense cases, advancing enumeration techniques.

Findings

Asymptotic formulas for sparse matrices with maximum degree o(n^{1/2})

Asymptotic formulas for dense matrices with uniform degree distribution

Extension of enumeration to graphs with loops and no multiple edges

Abstract

Let d=(d_1,d_2,..., d_n) be a vector of non-negative integers. We study the number of symmetric 0-1 matrices whose row sum vector equals d. While previous work has focussed on the case of zero diagonal, we allow diagonal entries to equal 1. When forming the row sum, each diagonal entry is multiplied by a factor of D, where D is 1 or 2. The case D=1 corresponds to enumeration by the usual row sum of matrices. The case D=2 corresponds to enumeration by degree sequence of undirected graphs with loops but no repeated edges, due to the convention that a loop contributes 2 to the degree of its incident vertex. We obtain asymptotically precise formulae for the number of matrices in the sparse range (where, roughly, the maximum row sum is o(n^{1/2})), and in the dense range (where, roughly, the average row sum is proportional to n and the row sums do not vary greatly).