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Let $G$ be a graph with adjacency matrix $A$. Let $s(G)$ be the number of connected components of $G$ that are cycles and $r(G)$ the number of connected components that are either $K_2$ or even cycles. Then $$\det(A) = \sum_{H} (-1)^{r(H)} 2^{s(H)}$$ where the sum is over all spanning subgraphs of $G$ that have only $K_2$ and cycles as their connected components.

In particular if $T$ is a tree the determinant of its adjacency matrix is $\pm$ the number of perfect matchings of $T$.

This identity can be generalized to all other coefficients of the characteristic polynomial of $A.$ For more information check the chapter "Determinant expansions" of Biggs' book on algebraic graph theory.

Let $G$ be a graph with adjacency matrix $A$. Let $s(G)$ be the number of connected components of $G$ that are cycles and $r(G)$ the number of connected components that are either $K_2$ or even cycles. Then $$\det(A) = \sum_{H} (-1)^{r(H)} 2^{s(H)}$$ where the sum is over all spanning subgraphs $H$ of $G$ that have only $K_2$ and cycles as their connected components.

In particular if $T$ is a tree the determinant of its adjacency matrix is $\pm$ the number of perfect matchings of $T$.

A relevant passage from N. Biggs: Algebraic Graph Theory. Second Edition. Cambridge University Press, is:

This identity can be generalized to all other coefficients of the characteristic polynomial of $A.$ For more information check the chapter "Determinant expansions" of Biggs' book on algebraic graph theory.

Let $G$ be a graph with adjacency matrix $A$. Let $s(G)$ be the number of connected components of $G$ that are cycles and $r(G)$ the number of connected components that are even cycles. Then $$\det(A) = \sum_{H} (-1)^{r(H)} 2^{s(H)}$$ where the sum is over all spanning subgraphs of $G$ that have only $K_2$ and cycles as their connected components.

In particular if $T$ is a tree the determinant of its adjacency matrix is $\pm$ the number of perfect matchings of $T$.

This identity can be generalized to all other coefficients of the characteristic polynomial of $A.$ For more information check the chapter "Determinant expansions" of Biggs' book on algebraic graph theory.

Let $G$ be a graph with adjacency matrix $A$. Let $s(G)$ be the number of connected components of $G$ that are cycles and $r(G)$ the number of connected components that are either $K_2$ or even cycles. Then $$\det(A) = \sum_{H} (-1)^{r(H)} 2^{s(H)}$$ where the sum is over all spanning subgraphs of $G$ that have only $K_2$ and cycles as their connected components.

In particular if $T$ is a tree the determinant of its adjacency matrix is $\pm$ the number of perfect matchings of $T$.

This identity can be generalized to all other coefficients of the characteristic polynomial of $A.$ For more information check the chapter "Determinant expansions" of Biggs' book on algebraic graph theory.

Let $G$ be a graph with adjacency matrix $A$. Let $s(G)$ be the number of connected components of $G$ that are cycles and $r(G)$ the number of connected components that are $K_2$. Then $$\det(A) = \sum_{H} (-1)^{r(H)} 2^{s(H)}$$ where the sum is over all spanning subgraphs of $G$ that have only $K_2$ and cycles as their connected components.

In particular if $T$ is a tree the determinant of its adjacency matrix is $\pm$ the number of perfect matchings of $T$.

This identity can be generalized to all other coefficients of the characteristic polynomial of $A.$ For more information check the chapter "Determinant expansions" of Biggs' book on algebraic graph theory.

Let $G$ be a graph with adjacency matrix $A$. Let $s(G)$ be the number of connected components of $G$ that are cycles and $r(G)$ the number of connected components that are even cycles. Then $$\det(A) = \sum_{H} (-1)^{r(H)} 2^{s(H)}$$ where the sum is over all spanning subgraphs of $G$ that have only $K_2$ and cycles as their connected components.

In particular if $T$ is a tree the determinant of its adjacency matrix is $\pm$ the number of perfect matchings of $T$.

This identity can be generalized to all other coefficients of the characteristic polynomial of $A.$ For more information check the chapter "Determinant expansions" of Biggs' book on algebraic graph theory.