What does this particular graph notation mean?

Question

I am trying to understand this paper about video summarization using a graph algorithm. Section 4.1 of the linked paper, describes two graphs, with the same node set, while the edges differ, as they have different defining equations for edge weight. And the paper later combines the two graphs using something like a dot product.

$G^\theta_{tc} = G.G^\theta$

What is this notation for graph description?

Answer 1

I'm not certain, but given that not long before it says

We negate values to transfer the difference into similarity and normalize matrix $G$

I suspect that it's a matrix multiplication, where the matrix is a weighted adjacency matrix.

Answer 2

Unfortunetaly their notation is not clear at all.

They say,

We negate values to transfer the difference into similarity and normalize matrix G

So it would seem like G is a matrix. Then they define,

We construct a graph G(V,W)

Which is already a bit strange because G(V,W) can denote a graph, and G some matrix. They are overloading G.

Then they drop the (V,W) with the G, and perform,

$ G^\theta_{tc} = G.G^\theta$

So $G$ is some matrix, and $G^\theta$ is some graph. I'm not even sure if the multiplication between a graph and matrix is well defined, unless they are multiplying some adjacency matrices together.