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9
votes
1answer
430 views

Has anyone seen this sort of graph property used before?

Consider the following property of a graph $G$: The graph $G$ has no independent cutset of vertices, $S$, such that the number of components of $G-S$ is more than $|S|$ (the size of $S$). (That is, ...
3
votes
1answer
493 views

Graphical representation of chromatic polynomial

Suppose $c(G,u)$ is chromatic polynomial of connected simple graph $G$. We know that $|c(G,-1)|$, as Stanley proved, is the total number of directed graph on $G$, without any cycle. Also, we know ...
1
vote
1answer
109 views

Non-alternating chromatic factors?

It is well-known that the coefficients of a chromatic polynomial alternate in sign. But is it possible for a chromatic polynomial to have a factor (over $\mathbb{Q}$) with coefficients which do not ...
3
votes
2answers
668 views

polynomials with the same discriminant

Let $p(x)$ be the chromatic polynomial of a special graph. Performing a certain type of operation on the graph changes $p$ by shifting it and adding a constant, say to: $q(x)=p(x+a) + b$. I have ...
7
votes
1answer
409 views

Graphs with the same chromatic symmetric function

Does anyone know more examples of two nonisomorphic connected graphs with the same chromatic symmetric function? The only pair I know is the one in Stanley's paper on c.s.f.'s ...
11
votes
0answers
990 views

Finding a chromatic polynomial by polynomial fitting

I would like to find the chromatic polynomial χ for the n by m rook's graph Gn,m for as many values of n and m possible. The rooks graph is also (a) the line graph of the complete bipartite graph ...