scholarly journals Counting Proper Colourings in 4-Regular Graphs via the Potts Model

10.37236/7743 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Ewan Davies
Keyword(s):  
Partition Function ◽  
Internal Energy ◽  
Lower Bounds ◽  
Regular Graph ◽  
Potts Model ◽  
Temperature Limit ◽  
Upper And Lower Bounds ◽  
Regular Graphs ◽  
Zero Temperature ◽  
First Case

We give tight upper and lower bounds on the internal energy per particle in the antiferromagnetic $q$-state Potts model on $4$-regular graphs, for $q\ge 5$. This proves the first case of a conjecture of the author, Perkins, Jenssen and Roberts, and implies tight bounds on the antiferromagnetic Potts partition function. The zero-temperature limit gives upper and lower bounds on the number of proper $q$-colourings of $4$-regular graphs, which almost proves the case $d=4$ of a conjecture of Galvin and Tetali. For any $q \ge 5$ we prove that the number of proper $q$-colourings of a $4$-regular graph is maximised by a union of $K_{4,4}$'s.  

scholarly journals Isoperimetric Numbers of Regular Graphs of High Degree with Applications to Arithmetic Riemann Surfaces

10.37236/651 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Dominic Lanphier ◽  
Jason Rosenhouse
Keyword(s):  
Large Class ◽  
Lower Bounds ◽  
Riemann Surfaces ◽  
Upper And Lower Bounds ◽  
Regular Graphs ◽  
Eigenvalue Spectrum ◽  
Asymptotic Bounds ◽  
High Degree

We derive upper and lower bounds on the isoperimetric numbers and bisection widths of a large class of regular graphs of high degree. Our methods are combinatorial and do not require a knowledge of the eigenvalue spectrum. We apply these bounds to random regular graphs of high degree and the Platonic graphs over the rings $\mathbb{Z}_n$. In the latter case we show that these graphs are generally non-Ramanujan for composite $n$ and we also give sharp asymptotic bounds for the isoperimetric numbers. We conclude by giving bounds on the Cheeger constants of arithmetic Riemann surfaces. For a large class of these surfaces these bounds are an improvement over the known asymptotic bounds.

scholarly journals On the Widom–Rowlinson Occupancy Fraction in Regular Graphs

2016 ◽  
Vol 26 (2) ◽  
pp. 183-194 ◽  
Author(s):  
EMMA COHEN ◽  
WILL PERKINS ◽  
PRASAD TETALI
Keyword(s):  
Partition Function ◽  
Regular Graph ◽  
Upper Bound ◽  
Complete Graphs ◽  
Regular Graphs ◽  
Interacting Particles ◽  
Special Case ◽  
Random Configuration

We consider the Widom–Rowlinson model of two types of interacting particles on d-regular graphs. We prove a tight upper bound on the occupancy fraction, the expected fraction of vertices occupied by a particle under a random configuration from the model. The upper bound is achieved uniquely by unions of complete graphs on d + 1 vertices, Kd+1. As a corollary we find that Kd+1 also maximizes the normalized partition function of the Widom–Rowlinson model over the class of d-regular graphs. A special case of this shows that the normalized number of homomorphisms from any d-regular graph G to the graph HWR, a path on three vertices with a loop on each vertex, is maximized by Kd+1. This proves a conjecture of Galvin.

scholarly journals On the Number of Matchings in Regular Graphs

10.37236/834 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
S. Friedland ◽  
E. Krop ◽  
K. Markström
Keyword(s):  
Lower Bounds ◽  
Degree Sequence ◽  
Bipartite Graphs ◽  
Expected Value ◽  
Upper And Lower Bounds ◽  
Regular Graphs

For the set of graphs with a given degree sequence, consisting of any number of $2's$ and $1's$, and its subset of bipartite graphs, we characterize the optimal graphs who maximize and minimize the number of $m$-matchings. We find the expected value of the number of $m$-matchings of $r$-regular bipartite graphs on $2n$ vertices with respect to the two standard measures. We state and discuss the conjectured upper and lower bounds for $m$-matchings in $r$-regular bipartite graphs on $2n$ vertices, and their asymptotic versions for infinite $r$-regular bipartite graphs. We prove these conjectures for $2$-regular bipartite graphs and for $m$-matchings with $m\le 4$.

ZERO-TEMPERATURE SKEWED CHIRAL POTTS MODEL

1994 ◽  
Vol 06 (05a) ◽  
pp. 869-885
Author(s):  
R. J. BAXTER
Keyword(s):  
Potts Model ◽  
Shift Operator ◽  
Temperature Limit ◽  
Zero Temperature ◽  
Chiral Potts Model ◽  
Functional Relations ◽  
Spurious Solutions ◽  
Intermediate Phases ◽  
Free Interfaces ◽  
Ice Model

Functional relations have previously been obtained for the eigenvalues of the transfer matrices of the chiral Potts model. Introducing skewed boundary conditions is equivalent to merely modifying the quantum number of the spin shift operator in the relations (which accounts for at least some of the previously noted "spurious solutions"). As a first step towards calculating the general interfacial tension, we consider the model in a zero-temperature limit. It is still non-trivial, there being near-vertical free interfaces separating domains of different spin value. These interfaces behave like the "lines of down arrows" in the ice model, so one may hope to follow Lieb and use the Bethe ansatz to evaluate the partition function. It turns out that this can indeed be done. There is no wetting of an interface by intermediate phases.

scholarly journals Improved Lower Bounds for the Orders of Even Girth Cages

10.37236/6015 ◽  
2016 ◽  
Vol 23 (3) ◽  
Author(s):  
Tatiana Baginová Jajcayová ◽  
Slobodan Filipovski ◽  
Robert Jajcay
Keyword(s):  
Lower Bound ◽  
Structural Property ◽  
Lower Bounds ◽  
Regular Graph ◽  
Regular Graphs ◽  
The Difference

The well-known Moore bound $M(k,g)$ serves as a universal lower bound for the order of $k$-regular graphs of girth $g$. The excess $e$ of a $k$-regular graph $G$ of girth $g$ and order $n$ is the difference between its order $n$ and the corresponding Moore bound, $e=n - M(k,g) $. We find infinite families of parameters $(k,g)$, $g$ even, for which we show that the excess of any $k$-regular graph of girth $g$ is larger than $4$. This yields new improved lower bounds on the order of $k$-regular graphs of girth $g$ of smallest possible order; the so-called $(k,g)$-cages. We also show that the excess of the smallest $k$-regular graphs of girth $g$ can be arbitrarily large for a restricted family of $(k,g)$-graphs satisfying a very natural additional structural property.

scholarly journals UPPER AND LOWER BOUNDS ON THE PARTITION FUNCTION OF THE HOFSTADTER MODEL

1996 ◽  
Vol 10 (09) ◽  
pp. 409-416
Author(s):  
ALEXANDER MOROZ
Keyword(s):  
Partition Function ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Unitary Equivalence ◽  
Arbitrary Integer ◽  
Translation Operators ◽  
Magnetic Translation ◽  
Derivatives Of

Using unitary equivalence of magnetic translation operators, explicit upper and lower convex bounds on the partition function of the Hofstadter model are given for any rational “flux” and any value of Bloch momenta. These bounds (i) generalise straightforwardly to the case of a general asymmetric hopping and to the case of hopping of the form [Formula: see text] with n arbitrary integer larger than or equal 2, and (ii) allow to derive bounds on the derivatives of the partition function.

scholarly journals A Threshold Phenomenon for Random Independent Sets in the Discrete Hypercube

2010 ◽  
Vol 20 (1) ◽  
pp. 27-51 ◽  
Author(s):  
DAVID GALVIN
Keyword(s):  
Partition Function ◽  
Long Range ◽  
Lower Bounds ◽  
Hard Core ◽  
Independent Set ◽  
Independent Sets ◽  
Upper And Lower Bounds ◽  
Specific Information ◽  
Sharp Transition ◽  
Threshold Phenomenon

Let I be an independent set drawn from the discrete d-dimensional hypercube Qd = {0, 1}d according to the hard-core distribution with parameter λ > 0 (that is, the distribution in which each independent set I is chosen with probability proportional to λ|I|). We show a sharp transition around λ = 1 in the appearance of I: for λ > 1, min{|I ∩ Ɛ|, |I ∩ |} = 0 asymptotically almost surely, where Ɛ and are the bipartition classes of Qd, whereas for λ < 1, min{|I ∩ Ɛ|, |I ∩ |} is asymptotically almost surely exponential in d. The transition occurs in an interval whose length is of order 1/d.A key step in the proof is an estimation of Zλ(Qd), the sum over independent sets in Qd with each set I given weight λ|I| (a.k.a. the hard-core partition function). We obtain the asymptotics of Zλ(Qd) for $\gl>\sqrt{2}-1$, and nearly matching upper and lower bounds for $\gl \leq \sqrt{2}-1$, extending work of Korshunov and Sapozhenko. These bounds allow us to read off some very specific information about the structure of an independent set drawn according to the hard-core distribution.We also derive a long-range influence result. For all fixed λ > 0, if I is chosen from the independent sets of Qd according to the hard-core distribution with parameter λ, conditioned on a particular v ∈ Ɛ being in I, then the probability that another vertex w is in I is o(1) for w ∈ but Ω(1) for w ∈ Ɛ.

scholarly journals On the Generalized Distance Energy of Graphs

Mathematics ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 17 ◽  
Author(s):  
Abdollah Alhevaz ◽  
Maryam Baghipur ◽  
Hilal A. Ganie ◽  
Yilun Shang
Keyword(s):  
Lower Bounds ◽  
Complete Graph ◽  
Connected Graph ◽  
Distance Matrix ◽  
Wiener Index ◽  
Diagonal Matrix ◽  
Upper And Lower Bounds ◽  
Star Graph ◽  
Extremal Graphs ◽  
Generalized Distance

The generalized distance matrix D α ( G ) of a connected graph G is defined as D α ( G ) = α T r ( G ) + ( 1 − α ) D ( G ) , where 0 ≤ α ≤ 1 , D ( G ) is the distance matrix and T r ( G ) is the diagonal matrix of the node transmissions. In this paper, we extend the concept of energy to the generalized distance matrix and define the generalized distance energy E D α ( G ) . Some new upper and lower bounds for the generalized distance energy E D α ( G ) of G are established based on parameters including the Wiener index W ( G ) and the transmission degrees. Extremal graphs attaining these bounds are identified. It is found that the complete graph has the minimum generalized distance energy among all connected graphs, while the minimum is attained by the star graph among trees of order n.

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