Infinite Family Of Graphs Research Articles

Arc-transitive regular cyclic covers of the complete bipartite graph $$\mathsf{K}_{p,p}$$ K p , p

Characterizing regular covers of edge-transitive or arc-transitive graphs is currently a hot topic in algebraic graph theory. In this paper, we will classify arc-transitive regular cyclic covers of the complete bipartite graph $$\mathsf{K}_{p,p}$$Kp,p for each odd prime $$p$$p. The classification consists of four infinite families of graphs. In particular, such covers exist for each odd prime $$p$$p. The regular elementary abelian covers of $$\mathsf{K}_{p,p}$$Kp,p are considered in a sequel.

Certificates for Properties of Stability Polynomials of Graphs

A stable (or independent) set is a set of vertices where no two of the vertices in the set are adjacent. The stability polynomial $A(G; p)$ of a graph $G$ is the probability that a set of randomly chosen vertices is stable where the probability of each vertex being chosen is $p$, with choices independent. This polynomial is analogous to the chromatic polynomial in a precise sense. This paper considers factorisation of stability polynomials, following work by Morgan and Farr on factorisation of the chromatic polynomial. The stability polynomial $A(G;p)$ is said to have an s-factorisation with s-factors $H_{1}$ and $H_{2}$ if $A(G; p) = A(H_{1};p) A(H_{2};p)$. This clearly occurs when $G$ is a disjoint union of $H_{1}$ and $H_{2}$. We find many other cases where such factorisation occurs even when $G$ is connected. We find 152 different s-factorisations of connected graphs of order at most 9, and two infinite families. We introduce certificates of s-factorisation to explain s-factorisations in terms of the structure of $G$. Short certificates for s-factorisations of connected graphs of order at most 6 are found. Upper bounds for the lengths of the certificates of s-factorisations are given. We also use certificates to explain stability equivalence, when two graphs have the same stability polynomial. We give certifications of stability equivalence for two infinite families of graphs.

An infinite family of graphs with a facile count of perfect matchings

Given a graph G=(V,E), let the triangulationG△=(V△,E△)ofG be the graph obtained from G by supplementing each uv∈E with a new vertex w along with new edges uw and wv (while retaining uv). Let dv be the degree of a vertex v∈V and let G be a tree T. Then it is proved that the count of perfect matchings of the Cartesian product of T△ with K2 is given as the product of factors dv+1 over all v∈V. Also under favorable conditions, the degree sequence of T△×K2 is reconstructed via factorization of the number of its perfect matchings. Previously introduced degree product polynomials play a helpful role.

A Decomposition Theorem for Binary Matroids with no Prism Minor

The prism graph is the dual of the complete graph on five vertices with an edge deleted, $K_5\backslash e$. In this paper we determine the class of binary matroids with no prism minor. The motivation for this problem is the 1963 result by Dirac where he identified the simple 3-connected graphs with no minor isomorphic to the prism graph. We prove that besides Dirac's infinite families of graphs and four infinite families of non-regular matroids determined by Oxley, there are only three possibilities for a matroid in this class: it is isomorphic to the dual of the generalized parallel connection of $F_7$ with itself across a triangle with an element of the triangle deleted; it's rank is bounded by 5; or it admits a non-minimal exact 3-separation induced by the 3-separation in $P_9$. Since the prism graph has rank 5, the class has to contain the binary projective geometries of rank 3 and 4, $F_7$ and $PG(3, 2)$, respectively. We show that there is just one rank 5 extremal matroid in the class. It has 17 elements and is an extension of $R_{10}$, the unique splitter for regular matroids. As a corollary, we obtain Dillon, Mayhew, and Royle's result identifying the binary internally 4-connected matroids with no prism minor [5].

Inflection points for network reliability

Given a finite, undirected graph G (possibly with multiple edges), we assume that the vertices are operational, but the edges are each independently operational with probability p. The (all-terminal) reliability, $\operatorname{Rel}(G,p)$ , of G is the probability that the spanning subgraph of operational edges is connected. It has been conjectured that reliability functions have at most one point of inflection in (0,1). We show that the all-terminal reliability of almost every simple graph of order n has a point of inflection, and there are indeed infinite families of graphs (both simple and otherwise) with more than one point of inflection.

Exclusivity structures and graph representatives of local complementation orbits

We describe a construction that maps any connected graph G on three or more vertices into a larger graph, H(G), whose independence number is strictly smaller than its Lovász number which is equal to its fractional packing number. The vertices of H(G) represent all possible events consistent with the stabilizer group of the graph state associated with G, and exclusive events are adjacent. Mathematically, the graph H(G) corresponds to the orbit of G under local complementation. Physically, the construction translates into graph-theoretic terms the connection between a graph state and a Bell inequality maximally violated by quantum mechanics. In the context of zero-error information theory, the construction suggests a protocol achieving the maximum rate of entanglement-assisted capacity, a quantum mechanical analogue of the Shannon capacity, for each H(G). The violation of the Bell inequality is expressed by the one-shot version of this capacity being strictly larger than the independence number. Finally, given the correspondence between graphs and exclusivity structures, we are able to compute the independence number for certain infinite families of graphs with the use of quantum non-locality, therefore highlighting an application of quantum theory in the proof of a purely combinatorial statement.

Galois groups of chromatic polynomials

AbstractThe chromatic polynomialP(G,λ) gives the number of ways a graphGcan be properly coloured in at mostλcolours. This polynomial has been extensively studied in both combinatorics and statistical physics, but there has been little work on its algebraic properties. This paper reports a systematic study of the Galois groups of chromatic polynomials. We give a summary of the Galois groups of all chromatic polynomials of strongly non-clique-separable graphs of order at most 10 and all chromatic polynomials of non-clique-separableθ-graphs of order at most 19. Most of these chromatic polynomials have symmetric Galois groups. We give five infinite families of graphs: one of these families has chromatic polynomials with a dihedral Galois group and two of these families have chromatic polynomials with cyclic Galois groups. This includes the first known infinite family of graphs that have chromatic polynomials with the cyclic Galois group of order 3.

On Families of Graphs of Large Cycle Indicator, Matrices of Large Order and Key Exchange Protocols With Nonlinear Polynomial Maps of Small Degree

This paper studies the group theoretical protocol of Diffie–Hellman key exchange in the case of symmetrical group \({S_{p^n}}\) and more general Cremona group \({C(\mathbb K^n)}\) of polynomial automorphisms of free module \({\mathbb K^n}\) over arbitrary commutative ring \({\mathbb K}\). This algorithm depends very much on the choice of the base \({g_n \in C( \mathbb K^n)}\). It is important to work with the base \({g_n \in C( \mathbb K^n)}\), which is a polynomial map of a small degree and a large order such that the degrees of all powers \({g_n^k}\) are also bounded by a small constant. We suggest fast algorithms for generation of a map \({g_n={f_n} \xi_nf_n^{-1}}\), where ξn is an affine transformation (degree is 1) of a large order and fn is a fixed nonlinear polynomial map in n variables such that \({f_n^{-1}}\) is also a polynomial map and both maps fn and \({f_n^{-1}}\) are of small degrees. The method is based on properties of infinite families of graphs with a large cycle indicator and families of graphs of a large girth in particular. It guaranties that the order of gn is tending to infinity as the dimension n tends to infinity. We propose methods of fast generation of special families of cubical maps fn such that \({f_n^{-1}}\) is also of degree 3 based on properties of families of graphs of a large girth and graphs with a large cycle indicator. At the end we discuss cryptographical applications of maps of the kind τ fn τ−1 and some graph theoretical problems motivated by such applications.

Lexicographic products with high reconstruction numbers

The reconstruction number of a graph is the smallest number of vertex-deleted subgraphs needed to uniquely determine the graph up to isomorphism. Bollobás showed that almost all graphs have reconstruction number equal to three. McMullen and Radziszowski published a catalogue of all graphs on at most ten vertices with reconstruction number greater than three. We introduce constructions that generalize the examples identified in their work. In particular, we use lexicographic products of vertex transitive graphs with certain starter graphs from the work of Myrvold and from the work of Harary and Plantholt to generate new infinite families of graphs with high reconstruction numbers. In the process, we settle a question of McMullen and Radziszowski.

TETRAVALENT ARC-TRANSITIVE GRAPHS WITH UNBOUNDED VERTEX-STABILIZERS

AbstractIt has long been known that there exist finite connected tetravalent arc-transitive graphs with arbitrarily large vertex-stabilizers. However, beside a well-known family of exceptional graphs, related to the lexicographic product of a cycle with an edgeless graph on two vertices, only a few such infinite families of graphs are known. In this paper, we present two more families of tetravalent arc-transitive graphs with large vertex-stabilizers, each significant for its own reason.

Periodic Graphs

Let $X$ be a graph on $n$ vertices with adjacency matrix $A$ and let $H(t)$ denote the matrix-valued function $\exp(iAt)$. If $u$ and $v$ are distinct vertices in $X$, we say perfect state transfer from $u$ to $v$ occurs if there is a time $\tau$ such that $|H(\tau)_{u,v}|=1$. If $u\in V(X)$ and there is a time $\sigma$ such that $|H(\sigma)_{u,u}|=1$, we say $X$ is periodic at $u$ with period $\sigma$. It is not difficult to show that if the ratio of distinct non-zero eigenvalues of $X$ is always rational, then $X$ is periodic. We show that the converse holds, from which it follows that a regular graph is periodic if and only if its eigenvalues are distinct. For a class of graphs $X$ including all vertex-transitive graphs we prove that, if perfect state transfer occurs at time $\tau$, then $H(\tau)$ is a scalar multiple of a permutation matrix of order two with no fixed points. Using certain Hadamard matrices, we construct a new infinite family of graphs on which perfect state transfer occurs.

Color Chain of a Graph

For a simple connected undirected graph G, $${\chi(G), \chi_f(G), \Psi_f(G), \phi(G), \partial \Gamma (G)}$$ and Ψ(G) denote respectively the chromatic number, fall chromatic number (assuming that it exists for G), fall achromatic number, b-chromatic number, partial Grundy number and achromatic number of G. It is shown in Dunbar et al. (J Combin Math & Combin Comput 33:257–273, 2000) that for any graph G for which fall coloring exists, $${\chi (G)\leq \chi_f(G) \leq \Psi_f (G) \leq \phi(G) \leq \partial \Gamma (G)\leq \Psi (G)}$$. In this paper, we exhibit an infinite family of graphs G with a strictly increasing color chain: $${\chi (G) < \chi_f(G) < \Psi_f (G) < \phi(G) < \partial \Gamma (G) < \Psi (G)}$$. The existence of such a chain was raised by Dunbar et al.

Lower bounds for the domination number and the total domination number of direct product graphs

A sharp lower bound for the domination number and the total domination number of the direct product of finitely many complete graphs is given: γ ( × i = 1 t K n i ) ≥ t + 1 , t ≥ 3 . Sharpness is established in the case when the factors are large enough in comparison to the number of factors. The main result gives a lower bound for the domination (and the total domination) number of the direct product of two arbitrary graphs: γ ( G × H ) ≥ γ ( G ) + γ ( H ) − 1 . Infinite families of graphs that attain the bound are presented. For these graphs it also holds that γ t ( G × H ) = γ ( G ) + γ ( H ) − 1 . Some additional parallels with the total domination number are made.

An Infinite Family of Graphs with the Same Ihara Zeta Function

In 2009, Cooper presented an infinite family of pairs of graphs which were conjectured to have the same Ihara zeta function. We give a proof of this result by using generating functions to establish a one-to-one correspondence between cycles of the same length without backtracking or tails in the graphs Cooper proposed. Our method is flexible enough that we are able to generalize Cooper's graphs, and we demonstrate additional families of pairs of graphs which share the same zeta function.

Space-efficient informational redundancy

We study the relation of autoreducibility and mitoticity for polylog-space many-one reductions and log-space many-one reductions. For polylog-space these notions coincide, while proving the same for log-space is out of reach. More precisely, we show the following results with respect to nontrivial sets and many-one reductions: 1. polylog-space autoreducible ⇔ polylog-space mitotic, 2. log-space mitotic ⇒ log-space autoreducible ⇒ ( log n ⋅ log log n ) -space mitotic, 3. relative to an oracle, log-space autoreducible ⇏ log-space mitotic. The oracle is an infinite family of graphs whose construction combines arguments from Ramsey theory and Kolmogorov complexity.

A Lower Bound for Network Navigability

In his seminal work, Kleinberg showed how to augment meshes using random edges, so that they become navigable; that is, greedy routing computes paths of polylogarithmic expected length between any pairs of nodes. This yields the crucial question of determining whether such an augmentation is possible for all graphs. In this paper, we answer this question negatively by exhibiting an infinite family of graphs that cannot be augmented to become navigable whatever the distribution of random edges is. Precisely, it was known that graphs of doubling dimension at most $O(\log\log n)$ are navigable. We show that for doubling dimension $\gg\log\log n$, an infinite family of graphs cannot be augmented to become navigable. Finally, we present a positive navigability result by studying the special case of square meshes of arbitrary dimension that we prove to always be augmentable to become navigable. This latter result complements Kleinberg's original result and shows that adding extra links can sometimes break the navigability.

Concentration on the Discrete Torus Using Transportation

The sub-Gaussian constant of a graph arises naturally in bounding the moment-generating function of Lipschitz functions on the graph, with a given probability measure on the set of vertices. The closely related spread constant of a graph measures the maximal variance over all Lipschitz functions on the graph. As such they are both useful (as demonstrated in the works of Bobkov, Houdré and Tetali and Alon, Boppana and Spencer) for describing the concentration of measure phenomenon in product graphs. An equivalent formulation of the sub-Gaussian constant using a transportation inequality, introduced by Bobkov and Götze, is investigated here in depth, leading to a new way of bounding the sub-Gaussian constant. A tight concentration result for the discrete torus is given as a concrete application. An infinite family of graphs is also provided here to demonstrate that, typically, the spread and the sub-Gaussian constants differ by an order of magnitude.

Cycle codes of graphs and MDS array codes

Abstract We investigate how to colour edges of graph so that any two colours make up a spanning tree. This problem is equivalent to transforming the cycle code of a graph into a Maximum Distance Separable (MDS) array code. Adopting this graph-theoretical interpretation allows us to provide a compact description of some families of low density MDS array codes in terms of cycle codes of coloured graphs. This includes a short description of Xu et al.'s “B-code”, together with its erasure and error decoding algorithms. We also give a partial answer to Xu et al.'s question about efficient error decoding for the dual B-code. We give alternative families of MDS array cycle codes, and in passing prove that an optimal MDS array cycle code of minimum column distance 4 is given by an appropriate colouring of the Petersen graph. We introduce double array colourings which allow the decoding of column or row errors and provides a new interesting graph theoretical problem. We give infinite families of graphs which admit double array colourings.

On an infinite family of graphs with information ratio 2 − 1/k

In this paper we consider the secret sharing problem on special access structures with minimal qualified subsets of size two, i.e. secret sharing on graphs. This means that the participants are the vertices of the graph and the qualified subsets are the subsets of V(G) spanning at least one edge. The information ratio of a graph G is denoted by R(G) and is defined as the ratio of the greatest size of the shares a vertex has to remember and of the size of the secret. Since the determination of the exact information ratio is a non-trivial problem even for small graphs (i.e. for V(G) = 6), every construction can be of particular interest. Let k be the maximal degree in G. In this paper we prove that R(G) = 2 − 1/k for every graph G with the following properties: (A) every vertex has at most one neighbour of degree one; (B) vertices of degree at least 3 are not connected by an edge; (C) the girth of the graph is at least 6. We prove this by using polyhedral combinatorics arguments and the entropy method.

Fair reception and Vizing's conjecture

AbstractIn this paper we introduce the concept of fair reception of a graph which is related to its domination number. We prove that all graphs G with a fair reception of size γ(G) satisfy Vizing's conjecture on the domination number of Cartesian product graphs, by which we extend the well‐known result of Barcalkin and German concerning decomposable graphs. Combining our concept with a result of Aharoni, Berger and Ziv, we obtain an alternative proof of the theorem of Aharoni and Szabó that chordal graphs satisfy Vizing's conjecture. A new infinite family of graphs that satisfy Vizing's conjecture is also presented. © 2009 Wiley Periodicals, Inc. J Graph Theory 61: 45‐54, 2009