Distance-regular Graph Of Diameter Research Articles

Further study of distance-regular graphs with classical parameters with b < −1

Let Γ be a distance-regular graph of diameter d with classical parameters (d,b,α,β) with b<−1. Van Dam, Koolen and Tanaka [Distance-regular graphs, Electron. J. Combin. (2016) DS22] surveyed the classification of Γ. From this survey, the following four cases have not been studied: (1) d≥3, c2=a1=1; (2) d≥3, c2=1, a2=a1>1; (3) d=3, c2=1, a2>a1>1; (4) d=3, c2>1, a1≥1. Here ai, bi, ci (0≤i≤d) are the intersection numbers of Γ. In this paper, we study the above four cases. Our main results are as follows. For the case (1), Γ is the triality graph ▪; for the case (2), Γ is the collinearity graph of a generalized hexagon of order (a1+1,(a1+1)3). In particular, if a1+1 is a prime power, then the classical parameters of Γ are realized by the triality graphs ▪; for the case (3), Γ does not exist; for the case (4), precisely one of the following (i)–(iv) holds: (i) Γ is the dual polar graph ▪, where a1+1 is a prime power; (ii) Γ is the extended ternary Golay code graph; (iii) Γ is the large Witt graph M24; (iv) Γ is the collinearity graph of a maximal regular near hexagon with classical parameters (d,b,α,β)=(3,−a1−1,−c2+a1a1,(a1+1)(c2+a1+1)) with a1≥2, 2≤c2≤(a1)2+a1+1. In particular, if 2≤a1≤107−1, then c2=2 and a1∈{3,4,7,10,17,22,31,52,157}.

Некоторые графы Шилла с b = 5 не существуют

A Shilla graph is a distance-regular graph of diameter 3 that has a second eigenvalue equal to a = a3 . Koolen and Park found admissible arrays of intersections of the Shill graphs with b = 3 (there were 12 of them). Belousov I.N. found feasible intersection arrays of the Shilla graphs with b = 4 (there were 50 of them) and b = 5 (there were 82 of them). It is proved in the paper that distance-regular Schill graphs with b = 5 and intersection arrays {305,248,62;1,2,244}, {315,256,64;1,2,252}, {345,280,64;1,4,276},{615,496,124;1,4,492}, {815,656,164;1,2,652}, {855,688,172;1,4,684}, {855,688,170;1,5,684}, {910,732,180;1,10, 728}, {1000,804,201;1,3,800}, {1045,840,210;1,6,836}, {1055,848,212;1,4,844}, {1080,868,215;1,5,864}, {1155,928,232;1,2,924}, {1185,952,245;1,5,948}, {1235,992,248;1,8,988},{1535,1232,308;1,8,1228}, {1560,1252,310;1,10,1248}, {1615,1296,324;1,12,1292}, {1665,1336,334;1,2,1332} do not exist.

DISTANCE-REGULAR GRAPH WITH INTERSECTION ARRAY {27, 20, 7; 1, 4, 21} DOES NOT EXIST

In the class of distance-regular graphs of diameter 3 there are 5 intersection arrays of graphs with at most 28 vertices and noninteger eigenvalue. These arrays are \(\{18,14,5;1,2,14\}\), \(\{18,15,9;1,1,10\}\), \(\{21,16,10;1,2,12\}\), \(\{24,21,3;1,3,18\}\), and \(\{27,20,7;1,4,21\}\). Automorphisms of graphs with intersection arrays \(\{18,15,9;1,1,10\}\) and \(\{24,21,3;1,3,18\}\) were found earlier by A.A. Makhnev and D.V. Paduchikh. In this paper, it is proved that a graph with the intersection array \(\{27,20,7;1,4,21\}\) does not exist.

D-magic strongly regular graphs

For a set of distances D, a graph G on n vertices is said to be D-magic if there exists a bijection and a constant k such that for any vertex x, where is the D-neighbourhood set of x. In this paper we utilize spectra of graphs to characterize strongly regular graphs which are D-magic, for all possible distance sets D. In addition, we provide necessary conditions for distance regular graphs of diameter 3 to be {1}-magic.

On the Cheeger constant for distance-regular graphs

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 λ12, where λ1 is the smallest positive eigenvalue of the Laplacian matrix. The subject of this paper is a conjecture of the authors that for distance-regular graphs the Cheeger constant is at most λ1. In particular, we prove the conjecture for the known infinite families of distance-regular graphs, distance-regular graphs of diameter 2 (the strongly regular graphs), several classes of distance-regular graphs with diameter 3, and most distance-regular graphs with small valency.

Codes in Shilla Distance-Regular Graphs

Let Γ be a distance-regular graph of diameter 3 containing a maximal 1-code C, which is locally regular and last subconstituent perfect. Then the graph Γ has intersection array {a(p + 1), cp, a + 1; 1,c, ap} or {a(p + 1), (a + 1)p, c;1, c, ap}, where a = a3, c = c2, and p = p 33 3 (Jurisic, Vidali). In the first case, Γ has eigenvalue θ2 = −1 and the graph Γ3 is pseudogeometric for GQ(p + 1, a). In the second case, Γ is a Shilla graph. We study Shilla graphs in which every two vertices at distance 2 belong to a maximal 1-code. It is proved that, in the case θ2 = −1, a graph with the specified property is either the Hamming graph H(3, 3) or a Johnson graph. We find necessary conditions for the existence of Q-polynomial Shilla graphs in which any two vertices at distance 3 lie in a maximal 1-code. In particular, we find two infinite families of feasible intersection arrays of Q-polynomial graphs with the specified property: {b(b2 − 3b)/2, (b − 2)(b − 1)2/2, (b − 2)t/2; 1, bt/2, (b2 − 3b)(b − 1)/2} (graphs with p 33 3 = 0) and {b2(b − 4)/2, (b2 − 4b + 2)(b − 1)/2, (b − 2)l/2; 1, bl/2, (b2 − 4b)(b − 1)/2} (graphs with p 33 3 = 1).

A new distance-regular graph of diameter 3 on 1024 vertices

The dodecacode is a nonlinear additive quaternary code of length 12. By puncturing it at any of the twelve coordinates, we obtain a uniformly packed code of distance 5. In particular, this latter code is completely regular but not completely transitive. Its coset graph is distance-regular of diameter three on 2^{10} vertices, with new intersection array {33,30,15;1,2,15}. The automorphism groups of the code, and of the graph, are determined. Connecting the vertices at distance two gives a strongly regular graph of (previously known) parameters (2^{10}, 495,238, 240). Another strongly regular graph with the same parameters is constructed on the codewords of the dual code. A non trivial completely regular binary code of length 33 is constructed.

Inverse Problems in Graph Theory: Nets

Let $$\varGamma $$ be a distance-regular graph of diameter 3 with strong regular graph $$\varGamma _3$$ . The determination of the parameters $$\varGamma _3$$ over the intersection array of the graph $$\varGamma $$ is a direct problem. Finding an intersection array of the graph $$\varGamma $$ with respect to the parameters $$\varGamma _3$$ is an inverse problem. Previously, inverse problems were solved for $$\varGamma _3$$ by Makhnev and Nirova. In this paper, we study the intersection arrays of distance-regular graph $$\varGamma $$ of diameter 3, for which the graph $${\bar{\varGamma }}_3$$ is a pseudo-geometric graph of the net $$PG_{m}(n, m)$$ . New infinite series of admissible intersection arrays for these graphs are found. We also investigate the automorphisms of distance-regular graph with the intersection array $$\{20,16,5; 1,1,16 \}$$ .

On Automorphisms of a Distance-Regular Graph with Intersection Array {69, 56, 10; 1, 14, 60}

Let Γ be a distance-regular graph of diameter 3 with eigenvalues θ0 > θ1 > θ2 > θ3. If θ2 = −1, then the graph Γ3 is strongly regular and the complementary graph \({\bar \Gamma _3}\) is pseudogeometric for pGc3(k, b1/c2). If Γ3 does not contain triangles and the number of its vertices v is less than 800, then Γ has intersection array {69, 56, 10; 1, 14, 60}. In this case Γ3 is a graph with parameters (392, 46, 0, 6) and \({\bar \Gamma _2}\) is a strongly regular graph with parameters (392, 115, 18, 40). Note that the neighborhood of any vertex in a graph with parameters (392, 115, 18, 40) is a strongly regular graph with parameters (115, 18, 1, 3) and its existence is unknown. In this paper, we find possible automorphisms of these strongly regular graphs and automorphisms of a hypothetical distance-regular graph with intersection array {69, 56, 10; 1, 14, 60}. In particular, it is proved that the latter graph is not ar-ctransitive.

Distance-Regular Shilla Graphs with b2 = c2

A Shilla graph is defined as a distance-regular graph of diameter 3 with second eigen-value θ1 equal to a3. For a Shilla graph, let us put a = a3 and b = k/a. It is proved in this paper that a Shilla graph with b2 = c2 and noninteger eigenvalues has the following intersection array: $$\left\{ {\frac{{{b^2}\left( {b - 1} \right)}}{2},\frac{{\left( {b - 1} \right)\left( {{b^2} - b + 2} \right)}}{2},\frac{{b\left( {b - 1} \right)}}{4};1,\frac{{b\left( {b - 1} \right)}}{4},\frac{{b{{\left( {b - 1} \right)}^2}}}{2}} \right\}$$ If Γ is a Q-polynomial Shilla graph with b2 = c2 and b = 2r, then the graph Γ has intersection array $$\left\{ {2tr\left( {2r + 1} \right),\left( {2r + 1} \right)\left( {2rt + t + 1} \right),r\left( {r + t} \right);1,r\left( {r + t} \right),t\left( {4{r^2} - 1} \right)} \right\}$$ and, for any vertex u in Γ, the subgraph Γ3(u) is an antipodal distance-regular graph with intersection array $$\left\{ {t\left( {2r + 1} \right),\left( {2r - 1} \right)\left( {t + 1} \right),1;1,t + 1,t\left( {2r + 1} \right)} \right\}$$ The Shilla graphs with b2 = c2 and b = 4 are also classified in the paper.

AUTOMORPHISMS OF DISTANCE-REGULAR GRAPH WITH INTERSECTION ARRAY {25; 16; 1; 1; 8; 25}

Makhnev and Samoilenko have found parameters of strongly regular graphs with no more than 1000 vertices, which may be neighborhoods of vertices in antipodal distance-regular graph of diameter 3 and with \(\lambda=\mu\). They proposed the program of investigation vertex-symmetric antipodal distance-regular graphs of diameter 3 with \(\lambda=\mu\), in which neighborhoods of vertices are strongly regular. In this paper we consider neighborhoods of vertices with parameters \((25,8,3,2)\).

On Algorithmic Problems Concerning Graphs of Higher Degree of Symmetry

Since the ancient determination of the five platonic solids the study of symmetry and regularity has always been one of the most fascinating aspects of mathematics. One intriguing phenomenon of studies in graph theory is the fact that quite often arithmetic regularity properties of a graph imply the existence of many symmetries, i.e. large automorphism group G. In some important special situation higher degree of regularity means that G is an automorphism group of finite geometry. For example, a glance through the list of distance regular graphs of diameter d < 3 reveals the fact that most of them are connected with classical Lie geometry. Theory of distance regular graphs is an important part of algebraic combinatorics and its applications such as coding theory, communication networks, and block design. An important tool for investigation of such graphs is their spectra, which is the set of eigenvalues of adjacency matrix of a graph. Let G be a finite simple group of Lie type and X be the set homogeneous elements of the associated geometry. The complexity of computing the adjacency matrices of a graph Gr on the vertices X such that Aut GR = G depends very much on the description of the geometry with which one starts. For example, we can represent the geometry as the totality of 1 cosets of parabolic subgroups 2 chains of embedded subspaces (case of linear groups), or totally isotropic subspaces (case of the remaining classical groups), 3 special subspaces of minimal module for G which are defined in terms of a G invariant multilinear form. The aim of this research is to develop an effective method for generation of graphs connected with classical geometry and evaluation of its spectra, which is the set of eigenvalues of adjacency matrix of a graph. The main approach is to avoid manual drawing and to calculate graph layout automatically according to its formal structure. This is a simple task in a case of a tree like graph with a strict hierarchy of entities but it becomes more complicated for graphs of geometrical nature. There are two main reasons for the investigations of spectra: (1) very often spectra carry much more useful information about the graph than a corresponding list of entities and relationships (2) graphs with special spectra, satisfying so called Ramanujan property or simply Ramanujan graphs (by name of Indian genius mathematician) are important for real life applications (see [13]). There is a motivated suspicion that among geometrical graphs one could find some new Ramanujan graphs.

On electric resistances for distance-regular graphs

We investigate the behavior of electric potentials on distance-regular graphs, and extend some results of a prior paper, Koolen and Markowsky (2010) [15]. Our main result shows that if the distance between points is measured by electric resistance then all points are close to being equidistant on a distance-regular graph with large valency. In particular, we show that the ratio between resistances between pairs of vertices in a distance-regular graph of diameter 3 or more is bounded by 1+6k, where k is the degree of the graph. We indicate further how this bound can be improved to 1+4k in most cases. A number of auxiliary results are also presented, including a discussion of the diameter 2 case as well as applications to random walks.

New families of completely regular codes and their corresponding distance regular coset graphs

In this paper three new infinite families of linear binary completely regular codes are constructed. They have covering radius ? = 3 and 4, and are halves of binary Hamming and binary extended Hamming codes of length n = 2 m ?1 and 2 m , where m is even. There are also shown some combinatorial (binomial) identities which are new, to our knowledge.These completely regular codes induce, in the usual way, i.e., as coset graphs, three infinite families of distance-regular graphs of diameter three and four. This description of such graphs is new.

On D. G. Higman's note on regular 3-graphs

We introduce the notion of a t -graph and prove that regular 3-graphs are equivalent to cyclic antipodal 3-fold covers of a complete graph. This generalizes the equivalence of regular two-graphs and Taylor graphs. As a consequence, an equivalence between cyclic antipodal distance regular graphs of diameter 3 and certain rank 6 commutative association schemes is proved. New examples of regular 3-graphs are presented.

Extremal 1-codes in distance-regular graphs of diameter 3

We study 1-codes in distance-regular graphs of diameter 3 that achieve three different bounds. We show that the intersection array of a distance-regular graph containing such a code has the form $${\{a(p+1), cp, a+1; 1, c, a p\}\quad{\rm or}\quad\{a(p+1), (a+1)p,c; 1, c, a p\}}$$ for c = c 2, a = a 3 and $${p = p_{33}^3}$$ . These two families contain 10 + 15 known feasible intersection arrays out of which four are uniquely realized by the Sylvester graph, the Hamming graph H(3, 3), the Doro graph and the Johnson graph J(9, 3), but not all members of these two families are feasible. We construct four new one-parameter infinite subfamilies of feasible intersection arrays, two of which have a nontrivial vanishing Krein parameter: $${\{(2r^2-1)(2r+1), 4r(r^2-1), 2r^2; 1, 2(r^2-1), r(4r^2-2)\}}$$ and $${\{2r^2(2r+1), (2r-1)(2r^2+r+1), 2r^2; 1, 2r^2, r(4r^2-1)\}}$$ for r > 1 (the second family actually generalizes to a two-parameter family with the same property). Using this information we calculate some triple intersection numbers for these two families to show that they must contain the desired code. Finally, we use some additional combinatorial arguments to prove nonexistence of distance-regular graphs with such intersection arrays.

Automorphisms of coverings of strongly regular graphs with parameters (81,20,1,6)

The possible orders and subgraphs of fixed points of the automorphisms of distance-regular graphs of diameter 4 which are r-coverings of a strongly regular graph with parameters (81, 20, 1, 6) for r ∈ {2, 3, 6} are found. It is proved that, if the automorphism group of a covering of the above type acts transitively on the set of vertices of the graph, then r = 3.

Almost 2-Homogeneous Graphs and Completely Regular Quadrangles

Many known distance-regular graphs have extra combinatorial regularities: One of them is t-homogeneity. A bipartite or almost bipartite distance-regular graph is 2-homogeneous if the number γ i = |{x | ∂(u, x) = ∂(v, x) = 1 and ∂(w, x) = i − 1}| (i = 2, 3,..., d) depends only on i whenever ∂(u, v) = 2 and ∂(u, w) = ∂(v, w) = i. K. Nomura gave a complete classification of bipartite and almost bipartite 2-homogeneous distance-regular graphs. In this paper, we generalize Nomura’s results by classifying 2-homogeneous triangle-free distance-regular graphs. As an application, we show that if Γ is a distance-regular graph of diameter at least four such that all quadrangles are completely regular then Γ is isomorphic to a binary Hamming graph, the folded graph of a binary Hamming graph or the coset graph of the extended binary Golay code of valency 24. We also consider the case Γ is a parallelogram-free distance-regular graph.

Strongly Closed Subgraphs in a Distance-Regular Graph with c 2 &gt; 1

Let Γ be a distance-regular graph of diameter d ≥ 3 with c 2 > 1. Let m be an integer with 1 ≤ m ≤ d − 1. We consider the following conditions: (SC) m : For any pair of vertices at distance m there exists a strongly closed subgraph of diameter m containing them. (BB) m : Let (x, y, z) be a triple of vertices with ∂ Γ (x, y) = 1 and ∂ Γ (x, z) = ∂ Γ (y, z) = m. Then B(x, z) = B(y, z). (CA) m : Let (x, y, z) be a triple of vertices with ∂ Γ (x, y) = 2, ∂ Γ (x, z) = ∂ Γ (y, z) = m and |C(z, x) ∩ C(z, y)| ≥ 2. Then C(x, z) ∪ A(x, z) = C(y, z) ∪ A(y, z). Suppose that the condition (SC) m holds. Then it has been known that the condition (BB) i holds for all i with 1 ≤ i ≤ m. Similarly we can show that the condition (CA) i holds for all i with 1 ≤ i ≤ m. In this paper we prove that if the conditions (BB) i and (CA) i hold for all i with 1 ≤ i ≤ m, then the condition (SC) m holds. Applying this result we give a sufficient condition for the existence of a dual polar graph as a strongly closed subgraph in Γ.

Pseudo 1-homogeneous distance-regular graphs

Let Γ be a distance-regular graph of diameter d≥2 and a 1≠0. Let θ be a real number. A pseudo cosine sequence for θ is a sequence of real numbers σ 0,…,σ d such that σ 0=1 and c i σ i−1+a i σ i +b i σ i+1=θ σ i for all i∈{0,…,d−1}. Furthermore, a pseudo primitive idempotent for θ is E θ =s ∑ d i=0 σ i A i , where s is any nonzero scalar. Let \(\hat{v}\) be the characteristic vector of a vertex v∈VΓ. For an edge xy of Γ and the characteristic vector w of the set of common neighbours of x and y, we say that the edge xy is tight with respect to θ whenever θ≠k and a nontrivial linear combination of vectors \(E\hat{x}\) , \(E\hat{y}\) and Ew is contained in \(\mathrm{Span}\{\hat{z}\mid z\in V{\Gamma},\ \partial(z,x)=d=\partial(z,y)\}\) . When an edge of Γ is tight with respect to two distinct real numbers, a parameterization with d+1 parameters of the members of the intersection array of Γ is given (using the pseudo cosines σ 1,…,σ d , and an auxiliary parameter ε). Let S be the set of all the vertices of Γ that are not at distance d from both vertices x and y that are adjacent. The graph Γ is pseudo 1-homogeneous with respect to xy whenever the distance partition of S corresponding to the distances from x and y is equitable in the subgraph induced on S. We show Γ is pseudo 1-homogeneous with respect to the edge xy if and only if the edge xy is tight with respect to two distinct real numbers. Finally, let us fix a vertex x of Γ. Then the graph Γ is pseudo 1-homogeneous with respect to any edge xy, and the local graph of x is connected if and only if there is the above parameterization with d+1 parameters σ 1,…,σ d ,ε and the local graph of x is strongly regular with nontrivial eigenvalues a 1 σ/(1+σ) and (σ 2−1)/(σ−σ 2).