Charles G PhD

This is an oriented graph for the Seymour Second Neighborhood Conjecture #SSNC. Nodes are in "rooted neighborhoods". We see patterns, number of nodes in each neighborhood. #MathIsBeautiful
1->7
2->6
3->5
Second, number of cycles.
1->1
2->2
3->2 (can't have 3)
alphaxiv.org/abs/2501.00614

This #ProofByContradiction is like building a house of cards. What we see is that: either the house will collapse, leading to a contradiction and producing that degree-doubling node, or we finish constructing the house, and every node fails to be a degree-doubling node. #math

What is this proof by contradiction?
We're assuming that the #SSNC is false. That means that every node in the graph has second neighborhood smaller than it's first neighborhood. Then we investigate the overall graph and see if it can still stand.

So what is this proof by contradiction?
It is a #BottomUpApproach. Starting with the #definitions. I realized that first and second neighbors are not good enough because you lose tracking info. Second neighbors do not know which first neighbor an edge came from. #research

I told the story about how excited I was with my initial results. The problem is, noone else was doing the problem #MyWay, which made collaboration difficult. This #KnowledgeGraph shows how the #SSNC problem has been #researched over the decades.

I'm excited to share my #Math paper, alphaxiv.org/abs/2501.00614
It is my constructive #ProofByContradicton of the Seymour Second Neighborhood Conjecture, using #DataStructures and #Algorithms, #BFS and #coding intuition and rigor to #Research this #OpenProblem in #GraphTheory.

This second expert saw the same problem with missing cases, but constructed a more abstract example.

That was 5 years ago. And that's where more #Research and the need for a #dataStructure came in. It combats these missing cases, but also brought order to the overall problem.

I tightened up my definitions and my proofs, so I thought. I understood why the missing cases were always resolved immediately in concrete examples.

I was so confident, I sent the paper for review to another expert. And ...He saw the same problem with these missing cases.

In the coming weeks I contacted him, showed him a very preliminary draft of a paper I was writing, and a JavaScript program I had written for this research.

The main point was I thought I had solved the problem.

The expert was concerned I had missed some cases.

An expert was giving a talk on combinatorial game theory - a fascinating talk. After the talk, I rush up to him to tell him I've been working on a 30 year old problem of his. I was sure I embarrassed myself, but he gave me his email. #ResearchJourney #AcademicLife #Mentorship

I'm excited to share my #Math paper, alphaxiv.org/abs/2501.00614
It is my constructive #ProofByContradicton of the Seymour Second Neighborhood Conjecture, using #DataStructures and #Algorithms, #BFS and #coding intuition and rigor to #Research this #OpenProblem in #GraphTheory.

Parent nodes have a burden: they do not want their degrees to double.
Children can assist by load balancing: forming cycles in the following neighborhoods.
The children also have this burden.
These dual burdens and load balancing shrink neighborhoods and yield a contradiction

Parent nodes have a burden: they do not want their degrees to double. Children can assist by load balancing: forming cycles in the following neighborhoods. The children also have this burden. These dual burdens and load balancing shrink neighborhoods and yield a contradiction

I'm excited to share my latest #Math paper, alphaxiv.org/abs/2501.00614
It is my constructive #ProofByContradicton of the Seymour Second Neighborhood Conjecture, using #DataStructures and #Algorithms, #BFS and #coding intuition and rigor to solve this #OpenProblem in #GraphTheory.

I'm excited to share my latest #Math paper, alphaxiv.org/abs/2501.00614
It is my constructive #ProofByContradicton of the Seymour Second Neighborhood Conjecture, using #DataStructures and #Algorithms, #BFS and #coding intuition and rigor to solve this #OpenProblem in #GraphTheory.

I'm excited to share my latest #Math paper, alphaxiv.org/abs/2501.00614
It is my constructive #ProofByContradicton of the Seymour Second Neighborhood Conjecture, using #DataStructures and #Algorithms, #BFS and #coding intuition and rigor to solve this #OpenProblem in #GraphTheory.

This image shows my early coding to try to solve the conjecture. What we see are The nodes of the same color in cycles, I called them interior neighbors. The nodes of different colors are I called exterior neighbors. Wanted to investigate this more. Not pictured here are back arcs.

An Algorithmic Approach to Finding Degree-Doubling Nodes in Oriented Graphs | alphaXiv View recent discussion. Abstract: The Seymour Second Neighborhood Conjecture (SSNC) claims that there will always exist a node whose out-degree doubles in the square of an oriented graph. In this pape...

I'm excited to share my latest #Math paper, alphaxiv.org/abs/2501.00614
It is my constructive #ProofByContradicton of the Seymour Second Neighborhood Conjecture, using #DataStructures and #Algorithms, #BFS and #coding intuition and rigor to solve this #OpenProblem in #GraphTheory.

My Biblical thought of the day, Jesus went through Samaria, a town that Jews would walk mikes to avoid. So after the Resurrection, did this continue? Did Christians like Paul continue looking for converts in Sameria?

Read Job. So was Satan.

As a person with ADHD I thought Pinterest would be the thing that saved my life because I could put all those screenshots somewhere. But it's so confusing.

A word cloud, with words of different sizes, and different colors, with purple for the largest, red medium, and green small. "Black" stands out prominently in purple. In red, we see "climate", "student", "network", "justice", "identity", "teacher", and "undergraduate".

Who bears the brunt of these cuts? Well, here's a word cloud of the most frequent terms from the titles and abstracts of terminated grants, with word size proportional to frequency:

Screenshot of table showing number of grants and total value terminated by NSF directorate. STEM Education directorate is highlighted, with 417 terminations totaling 322 million dollars. CSV of table: Directorate,No. Terminated,Total Value Biological Sciences,46,"$27,960,075" Computer and Information Science and Engineering,85,"$44,630,113" Engineering,94,"$33,472,485" Geosciences,52,"$36,454,740" Mathematical and Physical Sciences,30,"$6,418,312" Office of the Director,7,"$4,859,738" STEM Education,417,"$322,680,063" "Social, Behavioral and Economic Sciences",266,"$77,635,904" "Technology, Innovation and Partnerships",46,"$62,231,309"

UPDATE: We have data on >1000 terminated #NSF grants at grant-watch.us. Thanks to all who have been contributing data the past few days.

We've also added a "Reports" section with some analyses. For NSF, we see that the STEM education directorate has been absolutely pummeled.

men spell out SOS

Men kidnapped by the Trump regime spell out an SOS at the Bluebonnet ICE concentration camp.

The problem is that there are too many people like Donald Trump voting.

It's not just about caving. I think they (or those above them) partially believe in the orange man. We as a country do not support rights or freedom like we say we do. And we see that clear as day when those rights are given to people we don't like (like the far right feels about immigrants)

Take DOGE with you

A Pacers fan brought a free throw clock for Giannis 😂

Let's see if it stands

Another DEI hire?