The first question asked is the just the maximum leaf number of the graph. The problem of finding it is in general NP-Hard. I think a good one is this, which is algorithmic. A recent one is here. Note that the maximum leaf number is $n-d(G)$ where $d(G)$ is the connected domination number of the graph $G$.

By the way, your notation seems confusing. Not all vertices can have $d$ distinct neighbors if the graph is $d$-regular. The adjacent vertices always have one common neighbor, isnt it?

The first question asked is the just the maximum leaf number of the graph. The problem of finding it is in general NP-Hard. For references, I think a good one is this, which is algorithmic. A recent paper is here. Note that the maximum leaf number is $n-d(G)$ where $d(G)$ is the connected domination number of the graph $G$.

By the way, your notation seems confusing. Not all vertices can have $d$ distinct neighbors if the graph is $d$-regular. The adjacent vertices always have one common neighbor, isnt it?

The first question asked is the just the maximum leaf number of the graph. There are lot of papers in this area. I think a good one is this, which is algorithmic. A recent one is here. Note that the maximum leaf number is $n-d(G)$ where $d(G)$ is the connected domination number of the graph $G$.

By the way, your notation seems confusing. Not all vertices can have $d$ distinct neighbors if the graph is $d$-regular. The adjacent vertices always have one common neighbor, isnt it?

The first question asked is the just the maximum leaf number of the graph. The problem of finding it is in general NP-Hard. I think a good one is this, which is algorithmic. A recent one is here. Note that the maximum leaf number is $n-d(G)$ where $d(G)$ is the connected domination number of the graph $G$.

By the way, your notation seems confusing. Not all vertices can have $d$ distinct neighbors if the graph is $d$-regular. The adjacent vertices always have one common neighbor, isnt it?