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I am curious about the accuracy of the numbers quoted. In particular, are the Republican numbers inflated or Democratic numbers deflated?

The original source for all but the Trump numbers makes it clear that the people included in the count are

people who worked with the President before they ran, during their campaign, and after inauguration

...and that the numbers are for felony arrests (does not include misdemeanors).

Notability (original chart): seen on a prominent Facebook page, and a reverse image search turns up numerous articles and social media pages using it.

EDIT: Obviously this is not intended to be about anyone working for the executive branch, but senior members or presidential appointees.

Original chart, for those interested:

Wikipedia has a list of American federal politicians convicted of crimes. This list is broken down first by administration, and then by branch of government.

For the executive branch this list has:

• 
  

  Trump (1):

  
  
  
  1. 
     Michael Flynn
  
  


• 
  

  Obama (1):

  
  
  
  1. 
     David Petraeus
  
  

(Flynn was appointed by Obama as Director of the Defense Intelligence Agency, but Flynn is not separately list here)

• 
  

  G. W. Bush (9):

  
  
  

  1. J. Michael Farren

  
  

  2. Scott Bloch

  
  

  3. Felipe Sixto

  
  

  4. Robert E. Coughlin

  
  

  5. David Safavian

  
  

  6. Scooter Libby

  
  

  7. Lester Crawford

  
  

  8. Claude Allen

  
  

  9. John Korsmo

  
  


• 
  

  Clinton (2)

  
  
  

  1. Wade Sanders

  
  

  2. Darleen Druyun

  
  


• 
  

  G. H. W. Bush (1)

  
  
  
  1. 
     Catalina Vasquez Villalpando
  
  


• 
  

  Reagan (7)

  
  
  

  1. James G. Watt

  
  

  2. Deborah Gore Dean

  
  

  3. Michael Deaver

  
  

  4. James E. Gaines

  
  

  5. Victor D. Cohen

  
  

  6. Elliott Abrams

  
  



• Carter (0)



• 
  

  Ford (1)

  
  
  
  1. 
     Earl Butz
  
  


• 
  

  Nixon (11)

  
  
  

  1. Maurice Stans

  
  

  2. Spiro Agnew

  
  

  3. John N. Mitchell

  
  

  4. Richard Kleindienst

  
  

  5. H. R. Haldeman

  
  

  6. John Ehrlichman

  
  

  7. Egil Krogh

  
  

  8. John Dean

  
  

  9. Dwight Chapin

  
  

  10. Herbert W. Kalmbach

  
  

  11. Charles Colson

  
  

Nonetheless, if the entire executive branch is considered, there are multitude of other people that have been convicted.

The chart frames the discussion around different administrations; so the discussion is about what is unique to an administration (this is the only interpretation I can make in good-faith). The image draws attention to the fact that in recent history, there have been several administrations with much more criminal activity than others, and what these administrations have in common is that they are Republican.

Wikipedia has a quick summary of notable political scandals at https://en.wikipedia.org/wiki/List_of_federal_political_scandals_in_the_United_States . Here's a quick tally without finessing the data: count the number of bullet points for each administration. This overcounts by including data points that aren't in the OP image, like the Niger uranium forgeries (no one was indicted, arrested, or served jail time). Additionally, data is skewed towards the present (older administrations have fewer entries). But this should give a rough comparison by "degree of scandal" in a administration. Sub bullets on the list include notable individuals that were convicted or pardoned (e.g., in Watergate) so I've included a count of those as well.

Administration Party Bullets sub-bullets total
Barack Obama (2009–2017) D 8 2 10
George W. Bush (2001–2009) R 37 27 64
Bill Clinton (1993–2001) D 5 0 5
George H. W. Bush (1989–1993) R 3 6 9
Ronald Reagan (1981–1989) R 20 31 51
James E. Carter (1977–1981) D 1 0 1
Gerald Ford (1974–1977) R 1 0 1
Richard M. Nixon (1969–1974) R 8 16 24 

You can start debating items on the list, like how much of Iran-Contra should be under Reagan vs Bush. But I think this wikipedia list at least corroborates the general message that Nixon, Reagan, and W. Bush have had more political issues than other presidents.

But this wikipedia list isn't perfect. It doesn't include things like Nixon sacrificing 22,000 American lives to win the election (no one was indicted, arrested, or served jail time). Or the first US-ordered drone killing of a US citizen, under Obama (no one was indicted, arrested, or served jail time). I also came across an unmentioned incident under Clinton: "Former Housing and Urban Development Secretary Henry Cisneros was indicted by a federal grand jury" ( cnn ).

The book Corruption and American Politics highlights these scandals:

• Nixon: Watergate (in three incidents)


• Reagan: "the list of administration luminaries who were in ethical trouble is quite long"


• (no mention of H. Bush)


• Clinton: "serial affairs and dishonesty", "Starr Investigation"


• "a good case can be made that George W. Bush deserves to be ranked among the most corrupt in American history..." (but thinks it's still too soon in 2010 to give an accurate reflection of the presidency)

I'm not sure that Clinton should really be in the same category as Nixon and Reagan, but I wanted to include this perspective.

(Sorry for not being definitive in this conclusion, but that's my inner skeptic.) I think that the source for the image spends a lot of time discussing methodology and data sources, something a lot of internet meme posts don't do. At a high level, the methods and data are sound. Second, the original image is broadly supported (again, at a high level) by data from wikipedia. (Note: I am not citing a single statement as fact, but aggregating a set of statements; this should be more accurate). This broadly agrees with the book Corruption and American Politics, though perhaps the Clinton administration should be perceived as less pristine than initially portrayed in the source image.

Closing thoughts: It's true that Nixon, Reagan, and W. Bush had more political scandals than other presidencies; on the low end of "scandalness," are Clinton and H. Bush. I don't think the (R) next to their names is a sign of correlation to scandal though. For instance, Ford and H. Bush both had fewer problems than other Republican administrations.

No, the graphics are incorrect.

The graphic talks about "Executive Branch Criminal Activities". The Executive Branch is a well-defined term of the US Government, and includes many departments and government corporations.

Nidal Hasan was a member of the executive branch (in particular, the United States Army Medical Corps) during Barack Obama's presidential administration. He was convicted of 13 counts of murder and 32 counts of attempted murder for his November 5th, 2009 attack. He was sentenced to death.

However, the graphics suggest no members of the executive branch received indictments, convictions or sentences during the Obama administration.

Therefore, they are technically incorrect by official definitions.