At time $t=0$, let $G_n(V,E)$ be a graph with $n$ vertices and $m < n$ edges. Then there exists a unique symmetric adjacency matrix $A_n$ associated with $G_n(V,E)$, defined as follows: $a_{ij} = 1$ if there exists an edge between the vertex $i$ and vertex $j$, zero otherwise.

The spectrum, i.e., the set of eigenvalues of $A_n$ is not empty.

Then we consider at time $t=1$ a new vertex is added to the graph $G_n$ with probability $p$, $m<n+1$ edges connect the new vertex to the previous ones.

So the new adjacency matrix $A_{n+1}$ associated with the new graph $G_{n+1}$ has a new symmetric column/row where at least one entry is $1$ and in case all are ones.

Let us consider the sequence of matrices $G_n$ as $n$ tends to infinity. Can we say something on the associated sequence of eigenvalues?

Is there any reference on the subject?