Permanental sums of graphs of extreme sizes
Let G be a simple undirected graph. The permanental sum of G is equal to the permanent of the matrix I+A(G), where I is the identity matrix and A(G) is an adjacency matrix of G. The computation of permanental sum is #P-complete. In this paper, we compute the permanental sum of graphs of small sizes and derive explicit formulae for the permanental sum of graphs of large sizes. The results from small sizes are used as evidence to support a conjecture on an upper bound of the permanental sum of a graph in terms of its size. The results from large sizes are obtained by a new technique of employing rook's polynomial via the Principle of Inclusion and Exclusion.
National Natural Science Foundation of China
Extremal graph, Permanent, Permanental polynomial, Permanental sum
Mathematics and Statistics
Tingzeng Wu and Wasin So. "Permanental sums of graphs of extreme sizes" Discrete Mathematics (2021). https://doi.org/10.1016/j.disc.2021.112353