Counting independent sets in cubic graphs of given girth

We prove a tight upper bound on the independence polynomial (and total number of independent sets) of cubic graphs of girth at least 5. The bound is achieved by unions of the Heawood graph, the point/line incidence graph of the Fano plane. We also give a tight lower bound on the total number of independent sets of triangle-free cubic graphs. This bound is achieved by unions of the Petersen graph. We conjecture that in fact all Moore graphs are extremal for the scaled number of independent sets in regular graphs of a given minimum girth, maximizing this quantity if their girth is even and minimizing if odd. The Heawood and Petersen graphs are instances of this conjecture, along with complete graphs, complete bipartite graphs, and cycles.Postprint (author's final draft

Counting independent sets in cubic graphs of given girth

We prove a tight upper bound on the independence polynomial (and total number of independent sets) of cubic graphs of girth at least 5. The bound is achieved by unions of the Heawood graph, the point/line incidence graph of the Fano plane. We also give a tight lower bound on the total number of independent sets of triangle-free cubic graphs. This bound is achieved by unions of the Petersen graph. We conjecture that in fact all Moore graphs are extremal for the scaled number of independent sets in regular graphs of a given minimum girth, maximizing this quantity if their girth is even and minimizing if odd. The Heawood and Petersen graphs are instances of this conjecture, along with complete graphs, complete bipartite graphs, and cycles.Postprint (author's final draft

Fair Sets of Some Class of Graphs

Given a non empty set $S$ of vertices of a graph, the partiality of a vertex with respect to $S$ is the difference between maximum and minimum of the distances of the vertex to the vertices of $S$. The vertices with minimum partiality constitute the fair center of the set. Any vertex set which is the fair center of some set of vertices is called a fair set. In this paper we prove that the induced subgraph of any fair set is connected in the case of trees and characterise block graphs as the class of chordal graphs for which the induced subgraph of all fair sets are connected. The fair sets of $K_{n}$, $K_{m,n}$, $K_{n}-e$, wheel graphs, odd cycles and symmetric even graphs are identified. The fair sets of the Cartesian product graphs are also discussed.Comment: 14 pages, 4 figure

On Weak Odd Domination and Graph-based Quantum Secret Sharing

A weak odd dominated (WOD) set in a graph is a subset B of vertices for which there exists a distinct set of vertices C such that every vertex in B has an odd number of neighbors in C. We point out the connections of weak odd domination with odd domination, [sigma,rho]-domination, and perfect codes. We introduce bounds on \kappa(G), the maximum size of WOD sets of a graph G, and on \kappa'(G), the minimum size of non WOD sets of G. Moreover, we prove that the corresponding decision problems are NP-complete. The study of weak odd domination is mainly motivated by the design of graph-based quantum secret sharing protocols: a graph G of order n corresponds to a secret sharing protocol which threshold is \kappa_Q(G) = max(\kappa(G), n-\kappa'(G)). These graph-based protocols are very promising in terms of physical implementation, however all such graph-based protocols studied in the literature have quasi-unanimity thresholds (i.e. \kappa_Q(G)=n-o(n) where n is the order of the graph G underlying the protocol). In this paper, we show using probabilistic methods, the existence of graphs with smaller \kappa_Q (i.e. \kappa_Q(G)< 0.811n where n is the order of G). We also prove that deciding for a given graph G whether \kappa_Q(G)< k is NP-complete, which means that one cannot efficiently double check that a graph randomly generated has actually a \kappa_Q smaller than 0.811n.Comment: Subsumes arXiv:1109.6181: Optimal accessing and non-accessing structures for graph protocol

H-Colouring Bipartite Graphs

For graphs G and H, an H-colouring of G (or homomorphism from G to H) is a function from the vertices of G to the vertices of H that preserves adjacency. H-colourings generalize such graph theory notions as proper colourings and independent sets. For a given H, k∈V(H) and G we consider the proportion of vertices of G that get mapped to k in a uniformly chosen H-colouring of G. Our main result concerns this quantity when G is regular and bipartite. We find numbers 0⩽a−(k)⩽a+(k)⩽1 with the property that for all such G, with high probability the proportion is between a−(k) and a+(k), and we give examples where these extremes are achieved. For many H we have a−(k)=a+(k) for all k and so in these cases we obtain a quite precise description of the almost sure appearance of a randomly chosen H-colouring. As a corollary, we show that in a uniform proper q-colouring of a regular bipartite graph, if q is even then with high probability every colour appears on a proportion close to 1/q of the vertices, while if q is odd then with high probability every colour appears on at least a proportion close to 1/(q+1) of the vertices and at most a proportion close to 1/(q−1) of the vertices. Our results generalize to natural models of weighted H-colourings, and also to bipartite graphs which are sufficiently close to regular. As an application of this latter extension we describe the typical structure of H-colourings of graphs which are obtained from n-regular bipartite graphs by percolation, and we show that p=1/n is a threshold function across which the typical structure changes. The approach is through entropy, and extends work of J. Kahn, who considered the size of a randomly chosen independent set of a regular bipartite graph

On d-graceful labelings

In this paper we introduce a generalization of the well known concept of a graceful labeling. Given a graph G with e=dm edges, we call d-graceful labeling of G an injective function from V(G) to the set {0,1,2,..., d(m+1)-1} such that {|f(x)-f(y)| | [x,y]\in E(G)} ={1,2,3,...,d(m+1)-1}-{m+1,2(m+1),...,(d-1)(m+1)}. In the case of d=1 and of d=e we find the classical notion of a graceful labeling and of an odd graceful labeling, respectively. Also, we call d-graceful \alpha-labeling of a bipartite graph G a d-graceful labeling of G with the property that its maximum value on one of the two bipartite sets does not reach its minimum value on the other one. We show that these new concepts allow to obtain certain cyclic graph decompositions. We investigate the existence of d-graceful \alpha-labelings for several classes of bipartite graphs, completely solving the problem for paths and stars and giving partial results about cycles of even length and ladders.Comment: In press on Ars Combi

The Minimal Zn-Symmetric Graphs that are Not Zn-Spherical

Given a graph G equipped with faithful and fixed-point-free Γ-action (Γ a finite group) we define an orbit minor H of G to be a minor of G for which the deletion and contraction sets are closed under the Γ-action. The orbit minor H inherits a Γ-symmetry from G, and when the contraction set is acyclic the action inherited by H remains faithful and fixed-point free. When G embeds in the sphere and the Γ-action on G extends to a Γ-action on the entire sphere, we say that G is Γ-spherical. In this paper we determine for every odd value of n ≥ 3 the orbit-minor-minimal graphs G with a faithful and free Zn-action that are not Zn-spherical. There are 11 infinite families of such graphs, each of the 11 having exactly one member for each n. For n = 3, another such graph is K3,3. The remaining graphs are, essentially, the Cayley graphs for Zn aside from the cycle of length n. The result for n = 1 is exactly Wagner’s result from 1937 that the minor-minimal graphs that are not embeddable in the sphere are K5 and K3,3

Incidence geometry from an algebraic graph theory point of view

The goal of this thesis is to apply techniques from algebraic graph theory to finite incidence geometry. The incidence geometries under consideration include projective spaces, polar spaces and near polygons. These geometries give rise to one or more graphs. By use of eigenvalue techniques, we obtain results on these graphs and on their substructures that are regular or extremal in some sense. The first chapter introduces the basic notions of geometries, such as projective and polar spaces. In the second chapter, we introduce the necessary concepts from algebraic graph theory, such as association schemes and distance-regular graphs, and the main techniques, including the fundamental contributions by Delsarte. Chapter 3 deals with the Grassmann association schemes, or more geometrically: with the projective geometries. Several examples of interesting subsets are given, and we can easily derive completely combinatorial properties of them. Chapter 4 discusses the association schemes from classical finite polar spaces. One of the main applications is obtaining bounds for the size of substructures known as partial m- systems. In one specific case, where the partial m-systems are partial spreads in the polar space H(2d − 1, q^2) with d odd, the bound is new and even tight. A variant of the famous Erdős-Ko-Rado problem is considered in Chapter 5, where we study sets of pairwise non-trivially intersecting maximal totally isotropic subspaces in polar spaces. A combination of geometric and algebraic techniques is used to obtain a classification of such sets of maximum size, except for one specific polar space, namely H(2d − 1, q^2) for odd rank d ≥ 5. Near polygons, including generalized polygons and dual polar spaces, are studied in the last chapter. Several results on substructures in these geometries are given. An inequality of Higman on the parameters of generalized quadrangles is generalized. Finally, it is proved that in a specific dual polar space, a highly regular substructure would yield a distance- regular graph, generalizing a result on hemisystems. The appendix consists of an alternative proof for one of the main results in the thesis, a list of open problems and a summary in Dutch