Feynman diagrams, Feynman rules and corresponding integrals

Question

I would like some basic examples of Feynman diagrams: in particular I would like to understand how a Feynman diagram produces an integral: before I start let me made some remarks in the form of a

Disclaimer 0) I'm a mathematician, not a physicist
1) I'm interested in the theory where there is only one scalar field and the only interaction term in the Lagrangian is a monomial of the form $\frac{g}{d!}\varphi^d$-in particular for $d=3$ (non physical but nevermind) or $d=4$-here $g$ is a coupling constant
2) As I understood Feynman diagrams is a device to compute some complicated (functional) integrals-in particular I'm less interested in the physical interpretation of the diagrams: I'm only interested in an analytical information they contain.

Let me also specify concretely Feynman rules (fixing notations and conventions at the same time): 1) Every external line produces a propagator $\frac{1}{p_i^2+m^2}$
2) Every internal line produces an expression $\frac{1}{k_i^2+m^2} \frac{d^Dk_i}{(2 \pi)^D}$
3) Every vertex (corresponding to the monomial $\frac{g}{d!}\varphi^d$ produces delta distribution $(2\pi)^D g \delta(\sum_{lines entering}k_i-\sum_{lines living}k_j)$ (here we allow also external lines).
At the end we integrate the product of the all above expressions

For a Feyman diagram $\Gamma$ one can define the loop number $L$ as $I-V+1$ where $V$ is a number of vertices, $I$ is a number of internal lines. One can also define the so called superficial degree of summability as $DL-2I$: if this degree is negative then the integral associated to a graph will be finite. There is some subtelty that the loop number is undefined for graphs which are not connected: however the formula $I-V+1$ still makes sense but now this formula does not coincide with the number of loops of geometric realization of the graph.

These notions can be translated into the analytic language: the loop number should be the number of free variables over which (after we get rid of delta ,,functions'') we integrate. The superficial degree of summability comes from power counting: we integrate over $DL$ dimensional space and the expression under the integral is a rational function with a $2I$ degree polynomial in the denominator. However when I tried to evaluate some examples I met different expressions: since drawing Feynman diagrams is problematic I refer to this picture:

This picture contains several examples in which, by following Feynman rules I get expressions which look strange for me. I will be grateful if could somebody correct me and write the correct form of integrals corresponding to the Feynman graphs given in this image.

Answer 1

Well that's why the superficial degree of divergence is called superficial. If you look at say your first graph it has a bridge (the line in the middle of the dumbbell). In principle you should have a momentum there to be integrated over but conservation of momenta fixes it equal to $0$. This results in you being able to take the propagator out of the integral hence the factor $\frac{1}{m^2}$. The superficial degree of divergence counts this as a propagator as if it was still in the integral and thus gives $2D-6$. Your $2D-4$ corresponds to what one could call the "true" degree of divergence.