24 research outputs found
Unified spectral bounds on the chromatic number
One of the best known results in spectral graph theory is the following lower
bound on the chromatic number due to Alan Hoffman, where mu_1 and mu_n are
respectively the maximum and minimum eigenvalues of the adjacency matrix: chi
>= 1 + mu_1 / (- mu_n). We recently generalised this bound to include all
eigenvalues of the adjacency matrix.
In this paper, we further generalize these results to include all eigenvalues
of the adjacency, Laplacian and signless Laplacian matrices. The various known
bounds are also unified by considering the normalized adjacency matrix, and
examples are cited for which the new bounds outperform known bounds
New spectral bounds on the chromatic number encompassing all eigenvalues of the adjacency matrix
The purpose of this article is to improve existing lower bounds on the
chromatic number chi. Let mu_1,...,mu_n be the eigenvalues of the adjacency
matrix sorted in non-increasing order.
First, we prove the lower bound chi >= 1 + max_m {sum_{i=1}^m mu_i / -
sum_{i=1}^m mu_{n-i+1}} for m=1,...,n-1. This generalizes the Hoffman lower
bound which only involves the maximum and minimum eigenvalues, i.e., the case
. We provide several examples for which the new bound exceeds the {\sc
Hoffman} lower bound.
Second, we conjecture the lower bound chi >= 1 + S^+ / S^-, where S^+ and S^-
are the sums of the squares of positive and negative eigenvalues, respectively.
To corroborate this conjecture, we prove the weaker bound chi >= S^+/S^-. We
show that the conjectured lower bound is tight for several families of graphs.
We also performed various searches for a counter-example, but none was found.
Our proofs rely on a new technique of converting the adjacency matrix into
the zero matrix by conjugating with unitary matrices and use majorization of
spectra of self-adjoint matrices.
We also show that the above bounds are actually lower bounds on the
normalized orthogonal rank of a graph, which is always less than or equal to
the chromatic number. The normalized orthogonal rank is the minimum dimension
making it possible to assign vectors with entries of modulus one to the
vertices such that two such vectors are orthogonal if the corresponding
vertices are connected.
All these bounds are also valid when we replace the adjacency matrix A by W *
A where W is an arbitrary self-adjoint matrix and * denotes the Schur product,
that is, entrywise product of W and A
New measures of graph irregularity
In this paper, we define and compare four new measures of graph irregularity.
We use these measures to prove upper bounds for the chromatic number and the
Colin de Verdiere parameter. We also strengthen the concise Turan theorem for
irregular graphs and investigate to what extent Turan's theorem can be
similarly strengthened for generalized r-partite graphs. We conclude by
relating these new measures to the Randic index and using the measures to
devise new normalised indices of network heterogeneity
An inertial lower bound for the chromatic number of a graph
Let ) and denote the chromatic and fractional chromatic
numbers of a graph , and let denote the inertia of .
We prove that:
1 + \max\left(\frac{n^+}{n^-} , \frac{n^-}{n^+}\right) \le \chi(G) \mbox{
and conjecture that } 1 + \max\left(\frac{n^+}{n^-} , \frac{n^-}{n^+}\right)
\le \chi_f(G)
We investigate extremal graphs for these bounds and demonstrate that this
inertial bound is not a lower bound for the vector chromatic number. We
conclude with a discussion of asymmetry between and , including some
Nordhaus-Gaddum bounds for inertia