Please refer to the following articles
C. J. Liu and YutzeY. Chow. Enumeration of Connected Spanning Subgraphs Of A Planar GraphEnumeration of Connected Spanning Subgraphs Of A Planar Graph. Acta Mathematica Hungarica, 41(3):27–36, 1983. C
C. J. LIULiu and YUTZE CHOWYutze Chow. On Operator And Formal Sum Methods For Graph Enumeration ProblemsOn Operator And Formal Sum Methods For Graph Enumeration Problems. SIAM Journal of Control and Optimization, 5(3):384–406, 1984. C. J
Y. Liu and Yutze Chow. Enumeration of Connected Spanning Subgraphs Of A Planar GraphOn enumeration of spanning subgraphs with a preassigned cyclomatic number in a graph. Acta Mathematica Hungarica, 60(1):81–91, 1992. C
C. J. Liu and Yutze Chow. Enumeration Of Forests In GraphEnumeration of forests in a graph. Proceeding of American Mathematical Society, 83(3):659–662, 1981.