Sharp Bounds on the Permanental Sum of a Graph

Publication Date

11-1-2021

Document Type

Article

Publication Title

Graphs and Combinatorics

Volume

37

Issue

6

DOI

10.1007/s00373-021-02365-y

First Page

2423

Last Page

2437

Abstract

Let G be a simple undirected graph, I the identity matrix, and A(G) an adjacency matrix of G. Then the permanental sum of G equals to the permanent of the matrix I+ A(G). Since the computation of the permanental sum of a graph is #P-complete, it is desirable to have good bounds. In this paper, we affirm a sharp upper bound for general graphs conjectured by Wu and So. Moreover, we prove a sharp lower bound for connected tricyclic graphs. Lastly, several unsolved problems about permanental sum are presented.

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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