All Questions
18 questions
0
votes
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37
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Largest root of the Adjacency matrix of two graphs (comparison)
Let $G$ and $H$ be two graphs whose spectral radius (largest eigenvalue) of the adjacency matrix is the largest root of the following polynomial:
$$P_G(x) = x^6-x^5-(2a-n+5)x^4+(2a-n+1)x^3+2(5a-3n+5)x^...
- 199
2
votes
0
answers
123
views
Alon Tarsi reaches its lower bound for complete multipartite graphs
Consider a complete multipartite graph $G$ with $k$ $(k\ge2)$ parts with equal number of vertices $n$, where $n$ is even. If $k=2$, it is a well known result that the Alon-Tarsi number (ATN)(The ...
- 2,089
2
votes
1
answer
173
views
Matching polynomial, but $K_2$ is replaced by $K_3$. Have these been studied?
Given a simple graph $G=(V,E)$, we can consider matchings, $M\subseteq E$,
where $M$ is a matching iff no vertex is shared between different edges.
The number of edges in $M$ is denoted $|M|$.
The ...
- 15.8k
1
vote
0
answers
147
views
Counting Hamiltonian cycles in graph and finding a coefficient of polynomial
Exact result is #P-Hard, so we are looking for bounds.
Let $G$ be simple graph or simple digraph and $A$ its
adjacency matrix. $A$ is $n \times n$ with entries only zeros or ones.
Let $K=\mathbb{Z}[...
- 25.4k
3
votes
0
answers
203
views
Combinatorial characterizations of complex weight supports
This question is related to my last question and is originally motivated by recent advances in quantum physics.
I am looking for combinatorial characterizations of some algebraically defined families ...
- 5,409
4
votes
4
answers
305
views
Subgraph avoiding colorings
Let $P_{H}(G, t)$ be the number of vertex colorings of a graph $G$ in $t$ colors that avoid having a monochromatic subgraph $H$. In particular, for $H$ given by a single edge we recover the usual ...
- 645
-1
votes
1
answer
65
views
A follow-up question in a proof in a paper on complete multipartite graphs
A follow-up question from the following article/paper:
"Proof of a conjecture on distance energy change of complete multipartite graph due to edge deletion"
by Shaowei Sun and Kinkar Chandra ...
- 199
2
votes
0
answers
193
views
Possibly new multivariate polynomials associated to finite graphs
Let $G = (V,E)$ be a finite graph, where $V$ is a finite set of vertices and $E$ is a finite set of unoriented edges. We assume that $G$ has no loops, i.e. that there is no edge joining a vertex to ...
- 5,215
0
votes
0
answers
82
views
Proving Vizing's and Brooks' theorem using the polynomial approach
It is known that the graph polynomial defined by $\prod_{i<j}(x_i-x_j)$ where the vertices $x_k\ \ , \ \ k=\{1,2\ldots,n\}$ are ordered with respect to some order; can be used to verify the proper ...
- 2,089
2
votes
1
answer
304
views
Chromatic number and graph polynomial
If $\prod_{i=1}^t x_i^{e_i}$ is a monomial, define
$$rad\biggl(\prod_{i=1}^t x_i^{e_i}\biggr)$$
to be the number of distinct (nonzero) values of $e_i$.
Now let $G$ be a simple graph with vertices ...
- 2,089
2
votes
0
answers
52
views
The graph polynomial of the Total Graph of a Graph
Consider the Total Graph ($T(G)$) of a graph $G$ with vertex set $V$ edge set $E=\{(u,v)\}$ with Line graph $L(G)$ and subdivision graph $S(G)$ (formed by putting a vertex in each edge of the original ...
- 2,089
1
vote
1
answer
177
views
Refinement of face vectors of the simplicial noncrossing hypertree complexes of McCammond
Einziger on page 65 of "Incidence Hopf algebras: Antipodes, forest formulas, and noncrossing partitions" presents the antipode of a noncrossing partition Hopf algebra as a graded sequence of partition ...
- 10.5k
7
votes
0
answers
229
views
Has anyone seen these binary trees (Catalan-type related to the Gegenbauer polynomials and Motzkin paths)?
The OEIS entry A121448 enumerates binary trees with $n$ edges and $k$ vertices with outdegree 1.
Has anyone seen these trees?
The o.g.f. for this entry, $G(x,t)$, is essentially a discriminant ...
- 10.5k
22
votes
0
answers
550
views
Zero curves of Tutte Polynomials?
There is an extensive theory of the real and complex roots of the chromatic polynomial of a graph, a substantial fraction of this being due to the connections between the chromatic polynomial and a ...
- 12.7k
6
votes
0
answers
257
views
Are the roots of chromatic polynomials plus a fixed constant dense in $\mathbb{C}$?
Alan Sokal proved that chromatic roots are dense in the whole complex plane. I.e., if $P(G;z)$ denotes the chromatic polynomial of a finite simple graph $G$ evaluated at $z \in \mathbb{C}$, then $$\...
- 1,327
11
votes
1
answer
2k
views
Cycling through the Zeta Garden: Zeta functions for graphs, cycle index polynomials, and determinants
Zeta functions abound in mathematics. Audrey Terras describes in Zeta Functions and Chaos three zeta functions--the zeta fct. of a projective non-singular algebraic variety; the Artin-Mazur zeta ...
- 10.5k
13
votes
0
answers
385
views
Are the zeros of Tutte polynomials dense in $\mathbb C^2$?
For the chromatic polynomials of graphs we have two nice theorems which describe the behavior of their zeros: Thomassen proved that the set of real zeros of all chromatic polynomials is the union of $\...
- 85.6k
13
votes
5
answers
6k
views
Number of spanning forests in a graph
Hello,
I have two questions that have been bugging me recently. The first is about the number of spanning forests in a graph and the second is about enumerating these with edge labels.
Q1: I am ...
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