Sharp Bounds on the Permanental Sum of a Graph
Graphs and Combinatorics
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.
National Natural Science Foundation of China
Extremal graph, Permanent, Permanental polynomial, Permanental sum
Mathematics and Statistics
Wasin So, Tingzeng Wu, and Huazhong Lü. "Sharp Bounds on the Permanental Sum of a Graph" Graphs and Combinatorics (2021): 2423-2437. https://doi.org/10.1007/s00373-021-02365-y