Permanental sums of graphs of extreme sizes

Publication Date

6-1-2021

Document Type

Article

Publication Title

Discrete Mathematics

Volume

344

Issue

6

DOI

10.1016/j.disc.2021.112353

Abstract

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.

Funding Number

11761056

Funding Sponsor

National Natural Science Foundation of China

Keywords

Extremal graph, Permanent, Permanental polynomial, Permanental sum

Department

Mathematics and Statistics

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