It is known that $F(x)$ is the generating function of the counting sequence of connected simple graphs with N vertices is given by:

$F(x) = x + x^2 + 2x^3 + 6x^4 + 21x^5 + 112x^6 + 853x^7...$

where the total number of connected graphs of $n$ vertices is obtained by the coefficient corresponding to $x^n$.

The question here is: How to find the series (or formula) to get the counting of connected simple graphs with $n$ vertices of maximum degree up to $D$? (generalizing with at least $1$ vertex with maximum degree equal to $D$)

For example, for $n = 5$ with $D=3$, we have $8$ connected graphs with at least $1$ vertex with maximum degree equal to $3$, and for the same $n$ with for $D=4$, we have $11$ connected graphs with at least $1$ vertex with maximum degree equal to $4$, as shown by the distribution of degrees in the figure below:

Follows a table of the count of the first groups of simple graphs according to the number of vertices and the maximum degree: My question in brief is to determine the number of connected graphs with max. degree $D$ for a given number of Nodes $n$?

OBS: Do not get confused! the question does not include Disconnected Graphs, Digraphs, Labeled Graphs, number of cubic or quartic graphs!

Anyone know a solution to this intriguing problem? help me!

Note:

$F(x)$ is the generating function of the counting sequence of connected structures, then the corresponding generating function $G(x)$ of the counting sequence of all structures is given by:

$$G(x) = exp \sum_{k>=1} F(x^k)/k = \sum_{i=0} b_i x^k$$

Applying a variant of the Mobius inversion to equation above, it is possible to express $F(x)$ in terms of $G(x)$:

$$F(x) = exp \sum_{k>=1} [\mu(x)/k]log G(x^k) = \sum_{i=0} a_i x^k$$

Where $\mu$ stands for the Mobius function.

Let $a_n$ be the number of connected graphs on $n$ vertices and $b_n$ the number of all graphs on N vertices. Then:

$F(x) = x + x^2 + 2x^3 + 6x^4 + 21x^5 + 112x^6 + 853x^7 + ...$

and

$G(x) = 1 + x + 2x^2 + 4x^3 + 11x^4 + 34x^5 + 156x^6 + ...$

I saw this on "Counting Disconnected Structures: Chemical Trees, Fullerenes, I-graphs" on the link: https://hrcak.srce.hr/file/4232 but I did not understand how I can use it in favor of my problem.

An observation:

If $T(n)$ represents the total of connected graphs of $n$ nodes and $D_p(n)$ represents the total of connected sub-graphs group with max. degree $P$, and knowing that $P_{max} = (n-1)$, we can write: $$ T(n) = D_1(n) + D_2(n) + D_3(n) + ... + D_{n-1}(n)$$

For $n>2$, we know that $D_1(n) = 0$ and $D_2(n) = 2$, so we have to: $$T(n) = 2 + D_3(n) + ... + D_{n-1}(n)$$

Note also that $D_{n-1}$ can be obtained by the coefficient of $x^{n-1}$ of $G(x)$!

But how to calculate $D_p(n)$?

OBS: Another way to attack the problem I posted on the link: https://mathoverflow.net/questions/354705/about-counting-the-number-of-graphs-by-the-maximum-degree-d.