Antipodal covering graphs

A graph $\text{Γ}$ of valency k with a group G of automorphisms may be studied via the action of G on the vertex set $\text{V}\text{Γ}$. If G acts transitively on $\text{V}\text{Γ}$, then the notions of primitivity and imprimitivity are meaningful. We consider a natural notion of “block system” for a general graph $\text{Γ}$, which allows us to derive a “quotient” graph Γ whose vertices correspond to the blocks. The ideas are applied to antipodal systems in antipodal graphs: in particular we prove that for an antipodal distance-regular graph, the block size r cannot exceed the valency k; we further show that an antipodal distance-regular graph with r = k is (i) a circuit graph, (ii) a complete bipartite graph, or (iii) a threefold covering of Tutte's trivalent eight-cage.     

Eigenvalue multiplicities of highly symmetric graphs

We find lower bounds on eigenvalue multiplicities for highly symmetric graphs. In particular we prove:Theorem 1. If Γ is distance-regular with valency k and girth g (g?4), and λ (λ≠±?k) is an eigenvalue of Γ, then the multiplicity of λ is at least

$\text{k(k?1)}{}^{\mathrm{<mtext>[</mtext><mtext>g</mtext><mtext>4</mtext><mtext>]?1</mtext>}}$

if g≡0 or 1 (mod 4),

$\text{2(k?1)}{}^{\mathrm{<mtext>[</mtext><mtext>g</mtext><mtext>4</mtext><mtext>]</mtext>}}$

if g≡2 or 3 (mod 4) where [ ] denotes integer part. Theorem 2. If the automorphism group of a regular graph Γ with girth g (g?4) and valency k acts transitively on s-arcs for some s, $\text{1?s?[}\text{1}\text{2}\text{g]}$, then the multiplicity of any eigenvalue λ (λ≠±?k) is at least

$\text{k(k?1)}{}^{\mathrm{<mtext>s</mtext><mtext>2?1</mtext>}}$

if s is even,

$\text{2(k?1)}{}^{\mathrm{<mtext>(</mtext><mtext>s?1)</mtext><mtext>2</mtext>}}$

if s is odd.     

On Subgraphs in Distance-Regular Graphs

Terwilliger [15] has given the diameter bound d (s – 1)(k – 1) + 1 for distance-regular graphs with girth 2s and valency k. We show that the only distance-regular graphs with even girth which reach this bound are the hypercubes and the doubled Odd graphs. Also we improve this bound for bipartite distance-regular graphs. Weichsel [17] conjectures that the only distance-regular subgraphs of a hypercube are the even polygons, the hypercubes and the doubled Odd graphs and proves this in the case of girth 4. We show that the only distance-regular subgraphs of a hypercube with girth 6 are the doubled Odd graphs. If the girth is equal to 8, then its valency is at most 12.     

Antipodal graphs of diameter three

Distance-regular graphs of diameter three are of three (almost distinct) kinds: primitive, bipartite, and antipodal. An antipodal graph of diameter three is just an r-fold covering of a complete graph K_k+1 for some r?k. Its intersection array and spectrum are determined by the parameters r, k together with the number c of 2-arcs joining any two vertices at distance two. Most such graphs have girth three. In this note we consider antipodal distance-regular graphs of diameter three and girth ? 4. If r=2, then the only graphs are “K_{k+1, k+1} minus a 1-factor.” We therefore assume r?3. The graphs with c=1 necessarily have r=k and were classified in lsqb3rsqb. We prove the inequality $\text{r?2>c}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ (Theorem 2), list the feasible parameter sets when c=2 or 3 (Corollary 1), and conclude that every 3-fold or 4-fold antipodal covering of a complete graph has girth three (Corollary 2).     

Distance-regular graphs with girth 3 or 4: I

We study the structure of a distance-regular graph Γ with girth 3 or 4. First, we find some relationships among the intersection numbers of Γ when Γ contains a cycle {u₁, u₂, u₃, u₄} with ?(u₁, u₃) = ?(u₂, u₄) = 2. These relationships imply the diameter d, valency k, and intersection numbers a₁ and c_d of Γ are related by $\text{d ≤}\text{(k + c}{}_{\mathrm{d}}\text{)}\text{(a}{}_1\text{+ 2)}$. Next, we show subgraphs induced by vertex neighbourhoods in distance-regular graphs where cycles mentioned above do not exist are related to certain strongly regular graphs.     

Twice Q-polynomial distance-regular graphs of diameter 4

It is known that a distance-regular graph with valency k at least three admits at most two Qpolynomial structures. We show that all distance-regular graphs with diameter four and valency at least three admitting two Q-polynomial structures are either dual bipartite or almost dual bipartite. By the work of Dickie(1995) this implies that any distance-regular graph with diameter d at least four and valency at least three admitting two Q-polynomial structures is, provided it is not a Hadamard graph, either the cube H(d, 2)with d even, the half cube 1/2H(2d + 1, 2), the folded cube?H(2d + 1, 2), or the dual polar graph on [2A2d-1(q)]with q 2 a prime power.     

The girth of a thin distance-regular graph

Let Γ be a distance-regular graph of diameterd≥3. For each vertexx of Γ, letT(x) denote the Terwilliger algebra for Γ with respect tox. An irreducibleT(x)-moduleW is said to bethin if dimE _i ^* (x)W≤1 for 0≤i≤d, whereE _i ^* (x) is theith dual idempotent for Γ with respect tox. The graph Γ isthin if for each vertexx of Γ, every irreducibleT(x)-module is thin. Aregular generalized quadrangle is a bipartite distance-regular graph with girth 8 and diameter 4. Our main results are as follows: Theorem. Let Γ=(X,R) be a distance-regular graph with diameter d≥3 and valency k≥3. Then the following are equivalent:

1. Γis a regular generalized quadrangle.
2. Γis thin and c ₃=1.

Corollary. Let Γ=(X,R) be a thin distance-regular graph with diameter d≥3 and valency k≥3. Then Γ has girth 3, 4, 6, or 8. Then girth of Γ is 8 exactly when Γ is a regular generalized quadrangle.     

Max-cut and extendability of matchings in distance-regular graphs

A connected graph $G$ of even order $v$ is called $t$-extendable if it contains a perfect matching, $t<v/2$ and any matching of $t$ edges is contained in some perfect matching. The extendability of $G$ is the maximum $t$ such that $G$ is $t$-extendable. Since its introduction by Plummer in the 1980s, this combinatorial parameter has been studied for many classes of interesting graphs. In 2005, Brouwer and Haemers proved that every distance-regular graph of even order is $1$-extendable and in 2014, Cioabă and Li showed that any connected strongly regular graph of even order is 3-extendable except for a small number of exceptions.In this paper, we extend and generalize these results. We prove that all distance-regular graphs with diameter $D\ge 3$ are 2-extendable and we also obtain several better lower bounds for the extendability of distance-regular graphs of valency $k\ge 3$ that depend on $k$, $\lambda$ and $\mu$, where $\lambda$ is the number of common neighbors of any two adjacent vertices and $\mu$ is the number of common neighbors of any two vertices in distance two. In many situations, we show that the extendability of a distance-regular graph with valency $k$ grows linearly in $k$. We conjecture that the extendability of a distance-regular graph of even order and valency $k$ is at least $\lceil k/2\rceil -1$ and we prove this fact for bipartite distance-regular graphs.In course of this investigation, we obtain some new bounds for the max-cut and the independence number of distance-regular graphs in terms of their size and odd girth and we prove that our inequalities are incomparable with known eigenvalue bounds for these combinatorial parameters.     

Convex hull of the edges of a graph and near bipartite graphs

First we characterize the convex hull of the edges of a graph, edges viewed as the characteristic function of the hereditary closure of some subset of the 2-elements set of a finite set X. This characterization becomes more simple for a class of graphs that we call near bipartite, NBP in short. This class is then characterized as the class of graphs such that ?x?X, G_X\r(x), the induced subgraph of the complementary of the neighbourhood of x, is bipartite. We made a partial study of this class, whose interest is justified by the constatation that the following classes are strictly include: $\text{L(G)}$ the edge complementary of the line graph of G. NBP, $\text{K}{}_{13}$-free graphs.     

Distance-regular graphs admitting a perfect 1-code

In this paper, we study the problem that which of distance-regular graphs admit a perfect 1-code. Among other results, we characterize distance-regular line graphs which admit a perfect 1-code. Moreover, we characterize all known distance-regular graphs with small valency at most 4, the distance-regular graphs with known putative intersection arrays for valency 5, and all distance-regular graphs with girth 3 and valency 6 or 7 which admit a perfect 1-code.     

On maximal intersecting families of finite sets

Let r be a positive integer. A finite family $\text{H}$ of pairwise intersecting r-sets is a maximal clique of order r, if for any set A ? $\text{H}$, |A| ? r there exists a member E ? $\text{H}$ such that $\text{A ∩ E = ?}$. For instance, a finite projective plane of order r ? 1 is a maximal clique. Let N(r) denote the minimum number of sets in a maximal clique of order r. We prove $\text{N(r) ?}\text{3}\text{4}\text{r}{}^2$ whenever a projective plane of order $\text{r}\text{2}$ exists. This disproves the known conjecture N(r) ? r² ? r + 1.     

Extremal values of the interval number of a graph,II

It is shown that the interval number of a graph on n vertices is at most $\text{[}\text{1}\text{4}\text{(n+1)]}$, and this bound is best possible. This means that we can represent any graph on n vertices as an intersection graph in which the sets assigned to the vertices each consist of the union of at most $\text{[}\text{1}\text{4}\text{(n+1)]}$ finite closed intervals on the real line. This upper bound is attained by the complete bipartite graph K_[n/2],[n/2].     

Orientable and nonorientable genus of the complete bipartite graph

K_m,n is the complete bipartite graph with m and n vertices in its chromatic classes. G. Ringel has proved that the orientable genus of K_m,n is equal to $\text{{}\text{(m ? 2)(n ? 2)}\text{4}\text{}}\text{if}\text{m ≥ 2}\text{and}\text{n ≥ 2}$ and that its nonorientable genus is equal to $\text{{}\text{(m ? 2)(n ? 2)}\text{2}\text{}}\text{if}\text{m ≥ 3}\text{and}\text{n ≥ 3}$. We give new proofs of these results.     

Extremal non-bipartite regular graphs of girth 4

Let f(r) denote the smallest number of points in a non-bipartite r-regular graph of girth 4. It is known that $\text{f(r) ≥}\text{5r}\text{2}$ and that $\text{f(r) =}\text{5r}\text{2}$ if r is even. It is proved that $\text{f(r) ～}\text{5r}\text{2}$ and exact values for f(r) are provided for odd integers of the form r = 4n ? 1. Tight bounds for f(r) for odd integers of the form r= 4n + 1 are given.     

An intersection problem whose extremum is the finite projective space

Suppose that $\text{A}$ is a finite set-system of N elements with the property |A ∩ A′| = 0, 1 or k for any two different A, A′ ?A. We show that for N > k¹⁴

$\text{|a|=?}\text{N(N?1)(N?k)}\text{(k}{}^2\text{?k+1)(k}{}^2\text{?2k+1)}\text{+}\text{N(N?1)}\text{k(k?1)}\text{+N+1}$

where equality holds if and only if k = q + 1 (q is a prime power) $\text{N =}\text{(q}{}^{\mathrm{t+1}}\text{? 1)}\text{(q ? 1)}$ and $\text{A}$ is the set of subspaces of dimension at most two of the t-dimensional finite projective space of order q.     

Distance-regular graphs and (s,c, a,k)-graphs

A diameter-bound theorem for a class of distance-regular graphs which includes all those with even girth is presented. A new class of graphs, called (s, c, a, k)-graphs, is introduced, which are conjectured to contain enough of the local structure of finite distance-regular graphs for them all to be finite. It is proved that they are finite and a bound on the diameter is given in the case a ≤ c.     

Construction of matroidal families by partly closed sets

A matroidal family of graphs is a set $\text{M}$≠Ø of connected finite graphs such that for every finite graph G the edge sets of those subgraphs of G which are isomorphic to some element of $\text{M}$ are the circuits of a matroid on the edge set of G. In [9], Schmidt shows that, for n?0, ?2n<r?1, n, $\text{r∈}\text{Z}$, the set $\text{M}\text{(n, r)={G∣G}$ is a graph with β(G)=nα(G)+r and α(G )>, and is minimal with this property (with respect to the relation ?))} is a matroidal family of graphs. He also describes a method to construct new matroidal families of graphs by means of so-called partly closed sets. In this paper, an extension of this construction is given. By means of s-partly closed subsets of $\text{M}\text{(n, r)}$, s?r, we are able to give sufficient and necessary conditions for a subset $\text{P}\text{(n, r)}$ of $\text{M}\text{(n, r)}$ to yield a matroidal family of graphs when joined with the set $\text{I}\text{(n, s)}$ of all graphs $\text{G∈}\text{M}\text{(n, s)}$ which satisfy: If $\text{H∈}\text{P}\text{(n, r)}$, then H?G. In particular, it is shown that $\text{M}\text{(n, r)}$ is not a matroidal family of graphs for r?2. Furthermore, for n?0, 1?2n<r, n, $\text{r∈}\text{Z}$, the set of bipartite elements of $\text{M}\text{(n, r)}$ can be used to construct new matroidal families of graphs if and only if s?min(n+r, 1).     

On the minimum number of disjoint pairs in a family of finite sets

The following conjecture of Katona is proved. Let X be a finite set of cardinality n, 1 ? m ? 2ⁿ. Then there is a family $\text{F}$, |$\text{F}$| = m, such that F ∈ $\text{F}$, G ? X, | G | > | F | implies G ∈ $\text{F}$ and $\text{F}$ minimizes the number of pairs (F₁, F₂), F₁, F₂ ∈ $\text{F}$ F₁ ∩ F₂ = ? over all families consisting of m subsets of X.     

A property of matrices with positive determinants

Let $\text{P}$ be the set of all n × n real matrices which have a positive determinant. We show here that at least 2^{n ? 1} matrices are needed to “see” each matrix in $\text{P}$. Also, any finite subset of $\text{P}$ can be “seen” from a class of at most 2^{n ? 1} matrices in $\text{P}$.     

C1-algebras with finite duals

A forest is a finite partially ordered set F such that for x, y, z?F with x ? z, y ? z one has x ? y or y ? x. In this paper we give a complete characterization of all separable C¹-algebras $\text{A}$ with a finite dual $\text{A}\text{?}$, for which Prim $\text{A}$ is a forest with inclusion as partial order. These results are extended to certain separable C¹-algebras $\text{A}$ with a countable dual$\text{A}\text{?}$. As an example these results are used to characterize completely all separable C¹-algebras $\text{A}$ with a three point dual.