Jack Koolen

University of Science and Technology of China | USTC · School of Mathematical Sciences

In 2017, Qiao and Koolen showed that for any fixed integer $D\geqslant 3$, there are only finitely many such graphs with $\theta_{\min}\leqslant -\alpha k$, where $0<\alpha<1$ is any fixed number. In this paper, we will study non-bipartite distance-regular graphs with relatively small $\theta_{\min}$ compared with $k$. In particular, we will show t...

The tight-span of a finite metric space is a polytopal complex that has appeared in several areas of mathematics. In this paper we determine the polytopal structure of the tight-span of a totally split-decomposable (finite) metric. These metrics are a generalization of tree-metrics and have importance within phylogenetics. In previous work, we show...

In 2017, Qiao and Koolen showed that for any fixed integer $D\geq 3$, there are only finitely many such graphs with $\theta_{\min}\leq -\alpha k$, where $0<\alpha<1$ is any fixed number. In this paper, we will study non-bipartite distance-regular graphs with relatively small $\theta_{\min}$ compared with $k$. In particular, we will show that if $\t...

The Cheeger constant of a graph is the smallest possible ratio between the size of a subgraph and the size of its boundary. It is well known that this constant must be at least $\frac{\lambda_1}{2}$, where $\lambda_1$ is the smallest positive eigenvalue of the Laplacian matrix. The subject of this paper is a conjecture of the authors that for dista...

The regular complete t-partite graphs Kt×s (s,t positive integers at least 2) with valency k=(t−1)s have smallest eigenvalue −s=−k/(t−1), and hence, for fixed t there are infinitely many of them. In this paper we will show that these graphs are exceptional graphs for the class of distance-regular graphs. For this we will show a valency bound for di...

In this paper we give a new characterization of the dual polar graphs, extending the work of Brouwer and Wilbrink on regular near polygons. Also as a consequence of our characterization we confirm a conjecture of the authors on non-bipartite distance-regular graphs with smallest eigenvalue at most $-k/2$, where $k$ is the valency of the distance-re...

We determine the distance-regular graphs with diameter at least $3$ and $c_2\geq 2$ but without induced $K_{1,4}$-subgraphs.

We study regular graphs whose distance-$2$ graph or distance-$1$-or-$2$ graph is strongly regular. We provide a characterization of such graphs $\Gamma$ (among regular graphs with few distinct eigenvalues) in terms of the spectrum and the mean number of vertices at maximal distance $d$ from every vertex, where $d+1$ is the number of different eigen...

We consider nonregular graphs having precisely three distinct eigenvalues. The focus is mainly on the case of graphs having two distinct valencies and our results include the construction of new examples, structure theorems, parameter constraints, and a classification of certain special families of such graphs. We also present a new example of a gr...

We obtain several new results contributing to the theory of real equiangular line systems. Among other things, we present a new general lower bound on the maximum number of equiangular lines in d dimensional Euclidean space; we describe the two-graphs on 12 vertices; and we investigate Seidel matrices with exactly three distinct eigenvalues. As a r...

In this paper, we prove a number of related results on distance-regular graphs concerning electric resistance and simple random walk. We begin by proving several results on electric resistance; in particular we prove a sharp constant bounding the ratio of electrical resistances between any two pairs of points and give a counterexample to a conjectu...

The bilinear forms graph denoted here by $Bil_q(e\times d)$ is a graph defined on the set of $(e\times d)$-matrices ($e\geq d$) over $\mathbb{F}_q$ with two matrices being adjacent if and only if the rank of their difference equals $1$. In 1999, K. Metsch showed that the bilinear forms graph $Bil_q(e\times d)$ is characterized by its intersection a...

Juri\'{s}i\v{c} et al. conjectured that if a distance-regular graph $\Gamma$ with diameter $D$ at least three has a light tail, then one of the following holds: 1.$a_1 =0$; 2.$\Gamma$ is an antipodal cover of diameter three; 3.$\Gamma$ is tight; 4.$\Gamma$ is the halved $2D+1$-cube; 5.$\Gamma$ is a Hermitian dual polar graph $^2A_{2D-1}(r)$ where $...

Let $G$ be a distance-regular graph of order $v$ and size $e$. In this paper, we show that the max-cut in $G$ is at most $e(1-1/g)$, where $g$ is the odd girth of $G$. This result implies that the independence number of $G$ is at most $\frac{v}{2}(1-1/g)$. We use this fact to also study the extendability of matchings in distance-regular graphs. A g...

In this paper, we study the non-bipartite distance-regular graphs with valency $k$ and having a smallest eigenvalue at most $-k/2$.

We classify the connected graphs with precisely three distinct eigenvalues and second largest eigenvalue at most 1.

In this note, we construct bipartite 2-walk-regular graphs with exactly 6 distinct eigenvalues as incidence graphs of group-divisible designs with the dual property. For many of them, we show that they are 2-arc-transitive dihedrants. We note that many of these graphs are not described in Du et al. [7, Theorem1.2], in which they classify the connec...

From Alon and Boppana, and Serre, we know that for any given integer $k\geq 3$ and real number $\lambda<2\sqrt{k-1}$, there are finitely many $k$-regular graphs whose second largest eigenvalue is at most $\lambda$. In this paper, we investigate the largest number of vertices of such graphs.

This is a survey of distance-regular graphs. We present an introduction to distance-regular graphs for the reader who is unfamiliar with the subject, and then give an overview of some developments in the area of distance-regular graphs since the monograph `BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A., Distance-Regular Graphs, Springer-Verlag, Ber...

Dom de Caen posed the question whether connected graphs with three distinct eigenvalues have at most three distinct valencies. We do not answer this question, but instead construct connected graphs with four and five distinct eigenvalues and arbitrarily many distinct valencies. The graphs with four distinct eigenvalues come from regular two-graphs....

It is known that a distance-regular graph with valency k at least three admits at most two Q-polynomial 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 distan...

Let $\Gamma$ be a $Q$-polynomial distance-regular graph with diameter at least $3$. Terwilliger (1993) implicitly showed that there exists a polynomial, say $T(\lambda)\in \mathbb{C}[\lambda]$, of degree $4$ depending only on the intersection numbers of $\Gamma$ and such that $T(\eta)\geq 0$ holds for any non-principal eigenvalue $\eta$ of the loca...

Given a finite connected simple graph $G=(V,E)$ with vertex set $V$ and edge set $E\subseteq \binom{V}{2}$, we will show that $1.$ the (necessarily unique!) smallest block graph with vertex set $V$ whose edge set contains $E$ is uniquely determined by the $V$-indexed family ${\bf P}_G:=\big(\pi_0(G^{(v)})\big)_{v \in V}$ of the various partitions $...

In this paper we study distance-regular graphs with intersection array {(t + 1)s. ts. (t - 1)(s + 1 - psi); 1, 2, (t + 1)psi} (1) where s. t. psi are integers satisfying t >= 2 and 1 <= psi <= s. Geometric distance-regular graphs with diameter three and c(2) = 2 have such an intersection array. We first show that if a distance-regular graph with in...

In this paper, we show that the minimum number of vertices whose removal disconnects a connected strongly regular graph into non-singleton components, equals the size of the neighborhood of an edge for many graphs. These include blocks graphs of Steiner $2$-designs, many Latin square graphs and strongly regular graphs whose intersection parameters...

In this paper, we show that for given positive integer C, there are only finitely many distance-regular graphs with valency k at least three, diameter D at least six and k2k⩽C. This extends a conjecture of Bannai and Ito.

We give a structural classification of edge-signed graphs with smallest eigenvalue greater than -2. We prove a conjecture of Hoffman about the smallest eigenvalue of the line graph of a tree that was stated in the 1970s. Furthermore, we prove a more general result extending Hoffman's original statement to all edge-signed graphs with smallest eigenv...

In this note we study distance-regular graphs with a small number of vertices compared to the valency. We show that for a given α>2α>2, there are finitely many distance-regular graphs ΓΓ with valency kk, diameter D≥3D≥3 and vv vertices satisfying v≤αkv≤αk unless (D=3D=3 and ΓΓ is imprimitive) or (D=4D=4 and ΓΓ is antipodal and bipartite). We also s...

An important property of strongly regular graphs is that the second subconstituent of any primitive strongly regular graph is always connected. Brouwer asked to what extent this statement can be generalized to distance-regular graphs. In this paper, we show that if γ is any vertex of a distance-regular graph Γ and t is the index where the standard...

A $t$-walk-regular graph is a graph for which the number of walks of given length between two vertices depends only on the distance between these two vertices, as long as this distance is at most $t$. Such graphs generalize distance-regular graphs and $t$-arc-transitive graphs. In this paper, we will focus on 1- and in particular 2-walk-regular gra...

C. D. Godsil [Combinatorica 8, No. 4, 333–343 (1988; Zbl 0657.05047)] showed that if Γ is a distance-regular graph with diameter D≥3 and valency k≥3, and θ is an eigenvalue of Γ with multiplicity m≥2, then k≤(m+2)(m-1) 2. In this paper we will give a refined statement of this result. We show that if Γ is a distance-regular graph with diameter D≥3,...

We study the energy per vertex in regular graphs. For every k, we give an upper bound for the energy per vertex of a k-regular graph, and show that a graph attains the upper bound if and only if it is the disjoint union of incidence graphs of projective planes of order k-1 or, in case k=2, the disjoint union of triangles and hexagons. For every k,...

A split system on a finite set X is a set of bipartitions of X. Weakly compatible and k-compatible (k≥1) split systems are split systems which satisfy special restrictions on all subsets of a certain fixed size. They arise in various areas of applied mathematics such as phylogenetics and multi-commodity flow theory. In this note, we show that the n...

A realization of a finite metric space (X,d)(X,d) is a weighted graph (G,w)(G,w) whose vertex set contains XX such that the distances between the elements of XX in GG correspond to those given by dd. Such a realization is called optimal if it has minimal total edge weight. Optimal realizations have applications in fields such as phylogenetics, psyc...

In this paper, we study the distance-regular graphs Γ that have a pair of distinct vertices, say x and y, such that the number of common neighbors of x and y is about half the valency of Γ. We show that if the diameter is at least three, then such a graph, besides a finite number of exceptions, is a Taylor graph, bipartite with diameter three or a...

Let a connected graph Γ be locally disjoint union of at most three complete graphs and let Γ be cospectral with the Hamming graph H(3, q) (q ≥ 2). In this paper, we show that Γ is either the Hamming graph H(3, q) or the dual graph of H(3, 3) (i.e., the graph whose vertices are the triangles of H(3, 3) and two triangles are adjacent if they intersec...

A Shilla distance-regular graph Γ (say with valency k) is a distance-regular graph with diameter 3 such that its second largest eigenvalue equals to a 3. We will show that a 3 divides k for a Shilla distance-regular graph Γ, and for Γ we define b = b(Γ) := k a 3. In this paper we will show that there are finitely many Shilla distance-regular graphs...

We investigate fat Hoffman graphs with smallest eigenvalue at least -3, using their special graphs. We show that the special graph S(H) of an indecomposable fat Hoffman graph H is represented by the standard lattice or an irreducible root lattice. Moreover, we show that if the special graph admits an integral representation, that is, the lattice sp...

In this paper we will look at the relationship between the intersection number c2 and its diameter for a distance-regular graph. And also, we give some tools to show that a distance-regular graph with large c2 is bipartite, and a tool to show that if kD is too small then the distance-regular graph has to be antipodal.

We show that there is a disparity in fractal scaling behavior of the core and peripheral parts of empirical small-world scale-free networks. We decompose the network into a core and a periphery and measure the fractal dimension of each part separately using the box-counting method. We find that the core of small-world scale-free networks has a nonf...

A realisation of a metric d on a finite set X is a weighted graph (G,w) whose vertex set contains X such that the shortest-path distance between elements of X considered as vertices in G is equal to d. Such a realisation (G,w) is called optimal if the sum of its edge weights is minimal over all such realisations. Optimal realisations always exist,...

In this paper, we study a conjecture of Andries E. Brouwer from 1996 regarding the minimum number of vertices of a strongly regular graph whose removal disconnects the graph into non-singleton components. We show that strongly regular graphs constructed from copolar spaces and from the more general spaces called $\Delta$-spaces are counterexamples...

Let Γ be an antipodal distance-regular graph with diameter 4 and eigenvalues θ0>θ1>θ2>θ3>θ4. Then its Krein parameter vanishes precisely when Γ is tight in the sense of Jurišić, Koolen and Terwilliger, and furthermore, precisely when Γ is locally strongly regular with nontrivial eigenvalues p:=θ2 and −q:=θ3. When this is the case, the intersection...

We investigate the behavior of electric potentials on distance-regular graphs, and extend some results of a prior paper, J. Koolen and G. Markowsky [Electron. J. Comb. 17, No. 1, Research Paper R78, 15 p. (2010; Zbl 1225.05256)]. Our main result shows that if the distance between points is measured by electric resistance then all points are close t...

In this paper, we classify distance regular graphs such that all of its second largest local eigenvalues are at most one. Also we discuss the consequences for the smallest eigenvalue of a distance-regular graph. These extend a result by the first author, who classified the distance-regular graph with smallest eigenvalue $-1-\frac{b_1}{2}$.

Given a set $\Sg$ of bipartitions of some finite set $X$ of cardinality at least 2, one can associate to $\Sg$ a canonical $X$-labeled graph $\B(\Sg)$, called the Buneman graph. This graph has several interesting mathematical properties - for example, it is a median network and therefore an isometric subgraph of a hypercube. It is commonly used as...

In this paper, we show that for given positive integer C, there are only finitely many distance-regular graphs with valency k at least three, diameter D at least six and k2/k<=C. This extends a conjecture of Bannai and Ito. Comment: 12 pages

A Shilla distance-regular graph G (say with valency k) is a distance-regular graph with diameter 3 such that its second largest eigenvalue equals to a3. We will show that a3 divides k for a Shilla distance-regular graph G, and for G we define b=b(G):=k/a3. In this paper we will show that there are finitely many Shilla distance-regular graphs G with...

In this paper, we classify the connected non-bipartite integral graphs with spectral radius three.

Let G be a graph of order n such that \(\sum_{i=0}^{n}(-1)^{i}a_{i}\lambda^{n-i}\) and \(\sum_{i=0}^{n}(-1)^{i}b_{i}\lambda^{n-i}\) are the characteristic polynomials of the signless Laplacian and the Laplacian matrices of G, respectively. We show that a i ≥b i for i=0,1,…,n. As a consequence, we prove that for any α, 0<α≤1, if q 1,…,q n and μ 1,…,...

A non-complete geometric distance-regular graph is the point graph of a partial linear space in which the set of lines is a set of Delsarte cliques. In this paper, we prove that for a fixed integer m⩾2m⩾2, there are only finitely many non-geometric distance-regular graphs with smallest eigenvalue at least −m, diameter at least three and intersectio...

In this paper, we study distance-regular graphs $\Gamma$ that have a pair of distinct vertices, say x and y, such that the number of common neighbors of x and y is about half the valency of $\Gamma$. We show that if the diameter is at least three, then such a graph, besides a finite number of exceptions, is a Taylor graph, bipartite with diameter t...

We determine a lower bound for the spectral radius of a graph in terms of the number of vertices and the diameter of the graph. For the specific case of graphs with diameter three we give a slightly better bound. We also construct families of graphs with small spectral radius, thus obtaining asymptotic results showing that the bound is of the right...

For a distance-regular graph with second largest eigenvalue (resp. smallest eigenvalue) \mu1 (resp. \muD) we show that (\mu1+1)(\muD+1)<= -b1 holds, where equality only holds when the diameter equals two. Using this inequality we study distance-regular graphs with fixed second largest eigenvalue. Comment: 15 pages, this is submitted to Linear Algeb...

Let Γ be a Delsarte set graph with an intersection number c 2 (i.e., a distance-regular graph with a set C{\mathcal{C}} of Delsarte cliques such that each edge lies in a positive constant number nC{n_{\mathcal{C}}} of Delsarte cliques in C{\mathcal{C}}). We showed in Bang etal. (J Combin 28:501–506, 2007) that if ψ 1>1 then c 2 ≥ 2 ψ 1 where y1:=|G...

International audience In this paper, we explore completely regular codes in the Hamming graphs and related graphs. Experimental evidence suggests that many completely regular codes have the property that the eigenvalues of the code are in arithmetic progression. In order to better understand these "arithmetic completely regular codes", we focus on...

We first summarize the basic structure of the outer distribution module of a completely regular code. Then, employing a simple lemma concerning eigenvectors in association schemes, we propose to study the tightest case, where the indices of the eigenspace that appear in the outer distribution module are equally spaced. In addition to the arithmetic...

Let G be a graph on n vertices with r:=⌊n/2⌋ and let λ 1 ≥⋯≥λ n be adjacency eigenvalues of G. Then the Hückel energy of G, HE(G), is defined as HE(G)=2∑ i=1 r λ i ,ifn=2r;2∑ i=1 r λ i +λ r+1 ,ifn=2r+1· The concept of Hückel energy was introduced by Coulson as it gives a good approximation for the π-electron energy of molecular graphs. We obtain tw...

In their 1984 book "Algebraic Combinatorics I: Association Schemes", Bannai and Ito conjectured that there are only finitely many distance-regular graphs with fixed valency at least three. In a series of papers they showed that their conjecture holds for the class of bipartite distance-regular graphs. In this paper we prove that the Bannai-Ito conj...

Weakly compatible split systems are a generalization of unrooted evolutionary trees and are commonly used to display reticulate evolution or ambiguity in biological data. They are collections of bipartitions of a finite set X of taxa (e.g. species) with the property that, for every four taxa, at least one of the three bipartitions into two pairs (q...

In this paper we prove the Bannai-Ito conjecture, namely that there are only finitely many distance-regular graphs of fixed valency greater than two. Comment: 51 pages

Let $G$ be a graph on $n$ vertices with $r := \lfloor n/2 \rfloor$ and let $\lambda_1 \geq...\geq \lambda_{n} $ be adjacency eigenvalues of $G$. Then the H\"uckel energy of $G$, HE($G$), is defined as $$\he(G) = {ll} 2\sum_{i=1}^{r} \lambda_i, & \hbox{if $n= 2r$;} 2\sum_{i=1}^{r} \lambda_i + \lambda_{r+1}, & \hbox{if $n= 2r+1$.} $$ The concept of H...

Given a metric d on a finite set X, a realization of d is a triple (G,φ,w) consisting of a graph G=(V,E), a labeling φ:X→V, and a weighting w:E→R>0 such that for all x,y∈X the length of any shortest path in G between φ(x) and φ(y) equals d(x,y). Such a realization is called optimal if ‖G‖≔∑e∈Ew(e) is minimal amongst all realizations of d. In this p...

In this paper we will determine the spectra of the local graphs of the twisted Grassmann graphs and look at its consequences for the associated Terwilliger algebras. In particular, we show that the Terwilliger algebra for the twisted Grassmann graphs does depend on the base point. More specifically, we show that for the twisted Grassmann graphs the...

The vertex-connectivity of a distance-regular graph equals its valency.

In this note we show that there is a unique distance-regular graph with intersection array {280,243,144,10;1,8,90,280}.

We study graphs with spectral radius at most $\frac{3}{2}\sqrt{2}$ and refine results by Woo and Neumaier [On graphs whose spectral radius is bounded by $\frac{3}{2}\sqrt{2}$, Graphs Combinatorics 23 (2007), 713-726]. We study the limit points of the spectral radii of certain families of graphs, and apply the results to the problem of minimizing th...

Let Gamma be a triangle-free distance-regular graph with diameter d >= 3, valency k >= 3 and intersection number a(2) not equal 0. Assume Gamma has an eigenvalue with multiplicity k. We show that Gamma is 1-homogeneous in the sense of Nomura when d = 3 or when d >= 4 and a(4) = 0. In the latter case we prove that r is an antipodal cover of a strong...

Median graphs are a natural generalisation of trees and hypercubes that are closely related to distributive lattices and graph retracts. In the past decade, they have become of increasing interest to the biological community, where, amongst other things, they are applied to the study of evolutionary relationships within populations.Two simple measu...

In this paper, triangle-free distance-regular graphs with diameter 3 and an eigenvalue θ with multiplicity equal to their valency are studied. Let Γ be such a graph. We first show that θ=−1 if and only if Γ is antipodal. Then we assume that the graph Γ is primitive. We show that it is formally self-dual (and hence Q-polynomial and 1-homogeneous), a...

We study graphs with spectral radius at most frac(3, 2) sqrt(2) and refine results by Woo and Neumaier [R. Woo, A. Neumaier, On graphs whose spectral radius is bounded by frac(3, 2) sqrt(2), Graphs Combinatorics 23 (2007) 713-726]. We study the limit points of the spectral radii of certain families of graphs, and apply the results to the problem of...

AMS classifications: 05C50, 05E99;

Given a metric d on a finite set X, a realization of d is a weighted graph $G=(V,E,w\colon \ E \to {\Bbb R}_{>0})$ with $X \subseteq V$ such that for all $x,y \in X$ the length of any shortest path in G between x and y equals d(x,y). In this paper we consider two special kinds of realizations, optimal realizations and hereditarily optimal re...

Let Γ be an antipodal distance-regular graph of diameter 4, with eigenvalues $\theta_0>\theta_1>\theta_2>\theta_3>\theta_4$\theta_0>\theta_1>\theta_2>\theta_3>\theta_4. Then its Krein parameter q114q_{11}^4 vanishes precisely when Γ is tight in the sense of Jurišić, Koolen and Terwilliger, and furthermore, precisely when Γ is locally strongly regul...

We show that the Hamming graph H(3,q) with diameter three is uniquely determined by its spectrum for q⩾36. Moreover, we show that for given integer D⩾2, any graph cospectral with the Hamming graph H(D,q) is locally the disjoint union of D copies of the complete graph of size q-1, for q large enough.

We develop a theory of isometric subgraphs of hypercubes for which a certain inheritance of isometry plays a crucial role. It is well known that median graphs and closely related graphs embedded in hypercubes bear geometric features that involve realizations by solid cubical complexes or are expressed by Euler-type counting formula ef or cubical fa...

Let J be the all-ones matrix, and let A denote the adjacency matrix of a graph. An old result of Johnson and Newman states that if two graphs are cospectral with respect to yJ - A for two distinct values of y, then they are cospectral for all y. Here we will focus on graphs cospectral with respect to yJ - A for exactly one value widehat{y} of y. We...

The tight-span of a finite metric space is a polytopal complex with a structure that reflects properties of the metric. In this paper we consider the tight-span of a totally split-decomposable metric. Such metrics are used in the field of phylogenetic analysis, and a better knowledge of the structure of their tight-spans should ultimately provide i...

In this note we determine the full automorphism group of the twisted Grassmann graph. Further we show that twisted Grassmann graphs do not have antipodal distance-regular covers. At last, we show that the twisted Grassmann graphs are not the halved graphs of bipartite distance-regular graphs.

We classify triangle- and pentagon-free distance-regular graphs with diameter d⩾2d⩾2, valency k, and an eigenvalue multiplicity k. In particular, we prove that such a graph is isomorphic to a cycle, a k-cube, a complete bipartite graph minus a matching, a Hadamard graph, a distance-regular graph with intersection array {k,k-1,k-c,c,1;1,c,k-c,k-1,k}...

In this paper we study the absolute values of non-trivial eigenvalues of a distance-regular graph and find that these usually have large absolute value. We also give a motivation concerning a conjecture of Bannai and Ito.

In phylogenetic analysis, one searches for phylogenetic trees that reflect observed similarity between a collection of species in question. To this end, one often invokes two simple facts: (i) Any tree is completely determined by the metric it induces on its leaves (which represent the species). (ii) The resulting metrics are characterized by their...

Brouwer and Wilbrink showed that t + 1 • (s2 + 1)cd¡1 holds for a regular near 2d-gon of order (s,t) with s ‚ 2 and d is even. In this note we generalize their inequality to all diameter.

In phylogenetic analysis, one searches for phylogenetic trees that reflect observed similarity between a collection of species in question. To this end, one often invokes two simple facts: (i) Any tree is completely determined by the metric it induces on its leaves (which represent the species). (ii) The resulting metrics are characterized by their...

We show that the maximal number K2(n) of splits in a 2-compatible split system on an n-set is exactly 4n–10, for every n>3.Since K2(n)=CF3(n)/2 where CF3(n) is the maximal number of members in any 3-cross-free collection of (proper) subsets of an n-set, this gives a definitive answer to a question raised in 1979 by A. Karzanov who asked whether CF3...

Let Γ be a regular near polygon of order (s,t) with s>1 and t≥3. Let d be the diameter of Γ, and let r:= max{i∣(c i ,a i ,b i )=(c 1,a 1,b 1)}. In this note we prove several inequalities for Γ. In particular, we show that s is bounded from above by function in t if We also consider regular near polygons of order (s,3).

AMS classification: 05E30

In this note we classify the regular near polygons of order (s, 2).

In this paper we prove that there are finitely many triangle-free distance-regular graphs with degree 8, 9 or 10.

A spherical graph is a graph in which every interval is antipodal. Spherical graphs are an interesting generalization of hypercubes (a graph G is a hypercube if and only if G is spherical and bipartite). Besides hypercubes, there are many interesting examples of spherical graphs that appear in design theory, coding theory and geometry e.g., the Joh...

In this note we will generalize the Higman-Haemers inequalities for generalized polygons to thick regular near polygons.

In this paper we give a bound for the number ℓ(c,a,b) of columns (c,a,b)T in the intersection array of a distance-regular graph. We also show that this bound is intimately related to the Bannai–Ito conjecture.

Let be a graph with diameter d 2. Recall is 1-homogeneous (in the sense of Nomura) whenever for every edge xy of the distance partition{{z V() | (z, y) = i, (x, z) = j} | 0 i, j d}is equitable and its parameters do not depend on the edge xy. Let be 1-homogeneous. Then is distance-regular and also locally strongly regular with parameters (v,k,,), wh...

The width of a subset C of the vertices of a distance-regular graph is the maximum distance which occurs between elements of C. Dually, the dual width of a subset in a cometric association scheme is the index of the “last” eigenspace in the Q-polynomial ordering to which the characteristic vector of C is not orthogonal. Elementary bounds are derive...