In graph theory, we work with adjacency matrices which define the connections between the vertices. These matrices have various linear-algebraic properties. For example, their trace can be calculated (it is zero in the case of a loopless graph, i.e., an irreflexive symmetric binary relation). And we can also calculate their determinants.
How would you interpret the determinant in the context of a graph? For example, I teach network theory and the calculation of 'eigenvector centrality' requires the use of determinants. But the general question always comes up: what does the determinant mean in the context of the network (or graph)? Does it tell me of a property of the network that is useful?
In essence, I am trying to find a user-friendly interpretation of determinants in the context of networks or graphs. I would be grateful for any assistance.