What is the relation between linear subgraph and matching polynomial? I am confused about these following three concepts,


*

*An edge-cycle subgraph of a graph $G$ (also called a linear subgraph of $G$) is a subgraph of $G$ whose components are cycles and edges. 

*A set of subgraphs of $G$ whose components are all cycles. 

*A set of all $2-$regular subgraphs of $G$ 
Can someone help understand these definitions? May be give some examples? Are these three the same things? Is there a subset relation between these three? (Seems so!) 

Most importantly, I want to know how do these concepts help understand the matching polynomial. It seems that there is a way to rewrite the coefficients of the matching polynomial in terms of the above quantities. Can someone kindly help with that? 
Somehow coefficients of the characteristic polynomial can be written as sum over certain counting about this "edge-cycle subgraph". (..like the k^{th} coeffiecient there is related to the edge-cycle subgraphs which cover k vertices and  then then one of the terms in this sum is about matchings which cover k vertices...) Can someone give a reference to learn the proofs of this from? (or if you can type the proofs here incase that is something short!)
 A: One problem with your question is that it is not clear whether your various subgraphs are spanning subgraphs or not. If you are not assuming that they are spanning then a 2-regular subgraph is the same thing as a subgraph where each component is a cycle. As for the graphs in your first class, 
Biggs calls them sesquivalent in his book "Algebraic Graph Theory".
There is an expression for the coefficients of the characteristic polynomial of a graph in terms of the numbers of sesquivalent subgraphs. You will find this treated in Biggs's book, or in the book "Spectra of Graphs" by Cvetkovic, Doob and Sachs.
The formula is probably due to Harary, and I have heard it called the Harary-Sachs formula.
Guzman found an inclusion-exclusion type of identity relating the characteristic and matching polynomials of a graph. This is given as exercises in my "Algebraic Combinatorics"; the only other source I know for this is the original paper by me and Ivan Gutman "On the matching polynomial of a graph", which appeared in Algebraic Methods in Graph Theory, Vol. I, II (Szeged, 1978), volume 25 of Colloq. Math. Soc. Janos Bolyai, pages 241–249. North-Holland, Amsterdam, 1981. (This is not the most accessible of references, I admit.)
