3,662 research outputs found
Choice Number and Energy of Graphs
The energy of a graph G, denoted by E(G), is defined as the sum of the
absolute values of all eigenvalues of G. It is proved that E(G)>=
2(n-\chi(\bar{G}))>= 2(ch(G)-1) for every graph G of order n, and that E(G)>=
2ch(G) for all graphs G except for those in a few specified families, where
\bar{G}, \chi(G), and ch(G) are the complement, the chromatic number, and the
choice number of G, respectively.Comment: to appear in Linear Algebra and its Application
Beyond graph energy: norms of graphs and matrices
In 1978 Gutman introduced the energy of a graph as the sum of the absolute
values of graph eigenvalues, and ever since then graph energy has been
intensively studied.
Since graph energy is the trace norm of the adjacency matrix, matrix norms
provide a natural background for its study. Thus, this paper surveys research
on matrix norms that aims to expand and advance the study of graph energy.
The focus is exclusively on the Ky Fan and the Schatten norms, both
generalizing and enriching the trace norm. As it turns out, the study of
extremal properties of these norms leads to numerous analytic problems with
deep roots in combinatorics.
The survey brings to the fore the exceptional role of Hadamard matrices,
conference matrices, and conference graphs in matrix norms. In addition, a vast
new matrix class is studied, a relaxation of symmetric Hadamard matrices.
The survey presents solutions to just a fraction of a larger body of similar
problems bonding analysis to combinatorics. Thus, open problems and questions
are raised to outline topics for further investigation.Comment: 54 pages. V2 fixes many typos, and gives some new materia
Zero-free regions for multivariate Tutte polynomials (alias Potts-model partition functions) of graphs and matroids
The chromatic polynomial P_G(q) of a loopless graph G is known to be nonzero
(with explicitly known sign) on the intervals (-\infty,0), (0,1) and (1,32/27].
Analogous theorems hold for the flow polynomial of bridgeless graphs and for
the characteristic polynomial of loopless matroids. Here we exhibit all these
results as special cases of more general theorems on real zero-free regions of
the multivariate Tutte polynomial Z_G(q,v). The proofs are quite simple, and
employ deletion-contraction together with parallel and series reduction. In
particular, they shed light on the origin of the curious number 32/27.Comment: LaTeX2e, 49 pages, includes 5 Postscript figure
Exact Potts Model Partition Functions on Strips of the Honeycomb Lattice
We present exact calculations of the partition function of the -state
Potts model on (i) open, (ii) cyclic, and (iii) M\"obius strips of the
honeycomb (brick) lattice of width and arbitrarily great length. In the
infinite-length limit the thermodynamic properties are discussed. The
continuous locus of singularities of the free energy is determined in the
plane for fixed temperature and in the complex temperature plane for fixed
values. We also give exact calculations of the zero-temperature partition
function (chromatic polynomial) and , the exponent of the ground-state
entropy, for the Potts antiferromagnet for honeycomb strips of type (iv)
, cyclic, (v) , M\"obius, (vi) , cylindrical, and (vii)
, open. In the infinite-length limit we calculate and determine
the continuous locus of points where it is nonanalytic. We show that our exact
calculation of the entropy for the strip with cylindrical boundary
conditions provides an extremely accurate approximation, to a few parts in
for moderate values, to the entropy for the full 2D honeycomb
lattice (where the latter is determined by Monte Carlo measurements since no
exact analytic form is known).Comment: 48 pages, latex, with encapsulated postscript figure
From the Ising and Potts models to the general graph homomorphism polynomial
In this note we study some of the properties of the generating polynomial for
homomorphisms from a graph to at complete weighted graph on vertices. We
discuss how this polynomial relates to a long list of other well known graph
polynomials and the partition functions for different spin models, many of
which are specialisations of the homomorphism polynomial.
We also identify the smallest graphs which are not determined by their
homomorphism polynomials for and and compare this with the
corresponding minimal examples for the -polynomial, which generalizes the
well known Tutte-polynomal.Comment: V2. Extended versio
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