In graph theory, we work with adjacency matrices which define the connections between the vertices. These matrices have various properties in themselves. For example, their trace can be calculated (it is zero in the case of a non-recursive graph). 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 form a user-friendly interpretation of determinants in the context of networks or graphs. I would be grateful for any assistance.

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.