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### Are the roots of chromatic polynomials plus a fixed constant dense in $\mathbb{C}$?

Alan Sokal proved that chromatic roots are dense in the whole complex plane. I.e., if $P(G;z)$ denotes the chromatic polynomial of a finite simple graph $G$ evaluated at $z \in \mathbb{C}$, then ...

**9**

votes

**1**answer

443 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

**1**answer

577 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

**1**answer

114 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

**2**answers

730 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

**1**answer

450 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 ...

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**0**answers

1k 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 ...