How do we compare the nullity of an adjacency matrix and an incidence matrix of a simple graph?

Question

The nullity $\mu (G)$ of an adjacency matrix of a graph $G=(V, E)$ is defined as the multiplicity of the eigen value $0$. Moreover, $\mu (G)+r(G)=n$, where $r(G)$ and $n$ are rank and order, respectively of $G$. The nullity of an incidence matrix of $G$ is defined or denoted by $\mu (G)=|E|-n+k$, where $|E|$, $n $ and $k$ are size, order and components, respectively of $G$. Here, $\mu (G)+r(G)=|E|$. My observation: The above two definitions coincides when $n=|E|$, but nullity of an adjacency matrix doesn't depend on the components of the graph $G$, and the nullity of an incidence matrix cannot be be defined in terms of multiplicity of eigen value $0$. Can these two be compared in any other way better?

Answer 1

Short answer: No. As you say, the rank of the incidence matrix is determined by the number of edges, the number of vertices and the number of components. It is easy to find examples to show that these three parameters do not determine the rank of adjacency matrix, for example for trees the rank is the maximum number of vertices in a matching.

Although versions of this question does come up regularly, no useful formula for the rank of the adjacency matrix of a general graph is known.