I consider a connected simple finite graph $G$ and I define the equivalence relation $v\sim w$ on its vertex set such that $v\sim w$ if and only if $\mathrm{deg}(v)=\mathrm{deg}(w)$. I found that it is possible that the quotient graph of $G$ under this equivalence relation may be isomorph to the path graph of length $\delta$ where $\delta$ is the number of distinct degrees in $G$. Ordering the degrees by $d_1<...<d_{\delta}$ the existence of such an isomorphism implies that any vertex of degree $d_i$ in $G$ is only connected to vertices with degree $d_{i\pm 1}$.
But can I say more about $G$? Is there a general class of graphs which satisfies this property?
I am thankful for any input I can get. I am going to use it for Markov chains because I managed to prove that a certain Markov chain is reversible if and only if said isomorphism exists for the hereinbefore defined quotient graph.
Well, one thing we can do is ask for more structure. Write each $d_i$ as $a_i + b_i$ and ask that each vertex of degree $d_i$ is adjacent to $a_i$ vertices of degree $d_{i-1}$ and $b_i$ vertices of degree $d_{i+1}$. For the first degree $d_1$ we want $a_1 = 0$ and for the last degree $d_k$ we want $b_k = 0$; for the rest, we can choose $a_i$ and $b_i$ arbitrarily, but we want $a_i, b_i > 0$.
Then if there are $n_1, n_2, \dots, n_k$ vertices of degree $d_1 < d_2 < \dots < d_k$ respectively, we must have $n_i b_i = n_{i+1} a_{i+1}$ for the edge counts to work out, and we can do this just by having the proportion $n_1 : n_2 : \dots : n_k$ be right. We should scale up the proportion so that $n_i \ge \max\{b_{i-1}, a_{i+1}\}$ for all $i$, and then constructing the graph is possible...
...we just need a method for constructing a bipartite graph with $n_i$ vertices of degree $b_i$ on one side, and $n_{i+1}$ vertices of degree $a_{i-1}$ on the other side.
Here's one way. Just order the $n_i$ vertices on the first side cyclically. Give the first vertex on the other side edges to the first $a_{i-1}$ of them, then the next vertex edges to the next $a_{i-1}$, and so on, wrapping around when necessary. This will work out in the end. Actually, there are usually many ways to do this.
There are probably other graphs we can construct that don't have the regular $d_i = a_i + b_i$ split, but those are harder to find.