Aleksandar Jurišić

Tight Distance-Regular Graphs

AbstractWe consider a distance-regular graph Γ with diameter d ≥ 3 and eigenvalues k = θ0 > θ1 > ... > θd. We show the intersection numbers a1, b1 satisfy

1-Homogeneous Graphs with Cocktail Party μ-Graphs

\left( {\theta _1 + \frac{k}{{a_1 + 1}}} \right)\left( {\theta _d + \frac{k}{{a_1 + 1}}} \right) \geqslant - \frac{{ka_1 b_1 }}{{(a_1 + 1)^2 }}.

Using equality in the Krein conditions to prove nonexistence of certain distance-regular graphs

We say Γ is tight whenever Γ is not bipartite, and equality holds above. We characterize the tight property in a number of ways. For example, we show Γ is tight if and only if the intersection numbers are given by certain rational expressions involving d independent parameters. We show Γ is tight if and only if a1 ≠ 0, ad = 0, and Γ is 1-homogeneous in the sense of Nomura. We show Γ is tight if and only if each local graph is connected strongly-regular, with nontrivial eigenvalues −1 − b1(1 + θ1)−1 and −1 − b1(1 + θd)−1. Three infinite families and nine sporadic examples of tight distance-regular graphs are given.

Algorithmic Algebraic Combinatorics and Grbner Bases

We determine which Krein parameters of nonbipartite antipodal distance-regular graphs of diameter 3 and 4 can vanish, and give combinatorial interpretations of their vanishing. We also study tight distance-regular graphs of diameter 3 and 4. In the case of diameter 3, tight graphs are precisely the Taylor graphs. In the case of antipodal distance-regular graphs of diameter 4, tight graphs are precisely the graphs for which the Krein parameter q114 vanishes.

Triangle- and pentagon-free distance-regular graphs with an eigenvalue multiplicity equal to the valency

Let Γ be a graph with diameter d ≥ 2. Recall Γ is 1-homogeneous (in the sense of Nomura) whenever for every edge xy of Γ the distance partition{{z ∈ V(Γ) | ∂(z, y) = i, ∂(x, z) = j} | 0 ≤ i, j ≤ d}is equitable and its parameters do not depend on the edge xy. Let Γ be 1-homogeneous. Then Γ is distance-regular and also locally strongly regular with parameters (v′,k′,λ′,μ′), where v′ = k, k′ = a1, (v′ − k′ − 1)μ′ = k′(k′ − 1 − λ′) and c2 ≥ μ′ + 1, since a μ-graph is a regular graph with valency μ′. If c2 = μ′ + 1 and c2 ≠ 1, then Γ is a Terwilliger graph, i.e., all the μ-graphs of Γ are complete. In [11] we classified the Terwilliger 1-homogeneous graphs with c2 ≥ 2 and obtained that there are only three such examples. In this article we consider the case c2 = μ′ + 2 ≥ 3, i.e., the case when the μ-graphs of Γ are the Cocktail Party graphs, and obtain that either λ′ = 0, μ′ = 2 or Γ is one of the following graphs: (i) a Johnson graph J(2m, m) with m ≥ 2, (ii) a folded Johnson graph J¯(4m, 2m) with m ≥ 3, (iii) a halved m-cube with m ≥ 4, (iv) a folded halved (2m)-cube with m ≥ 5, (v) a Cocktail Party graph Km × 2 with m ≥ 3, (vi) the Schläfli graph, (vii) the Gosset graph.

Merging in antipodal distance-regular graphs

We find an inequality involving the eigenvalues of a regular graph; equality holds if and only if the graph is strongly regular. We apply this inequality to the first subconstituents of a distance-regular graph and obtain a simple proof of the fundamental bound for distance-regular graphs, discovered by Juri?i?, Koolen and Terwilliger. Using this we show that for distance-regular graphs with certain intersection arrays, the first subconstituent graphs are strongly regular. From these results we prove the nonexistence of distance-regular graphs associated to 20 feasible intersection arrays from the book Distance-Regular Graphs by Brouwer, Cohen and Neumaier .

AT4 family and 2-homogeneous graphs

We prove the nonexistence of a distance-regular graph with intersection array {74,54,15;1,9,60} and of distance-regular graphs with intersection arrays{4r^3+8r^2+6r+1,2r(r+1)(2r+1),2r^2+2r+1;1,2r(r+1),(2r+1)(2r^2+2r+1)} with r an integer and r>=1. Both cases serve to illustrate a technique which can help in determining structural properties for distance-regular graphs and association schemes with a sufficient number of vanishing Krein parameters.

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

Tutorials.- Loops, Latin Squares and Strongly Regular Graphs: An Algorithmic Approach via Algebraic Combinatorics.- Siamese Combinatorial Objects via Computer Algebra Experimentation.- Using Grobner Bases to Investigate Flag Algebras and Association Scheme Fusion.- Enumerating Set Orbits.- The 2-dimensional Jacobian Conjecture: A Computational Approach.- Research Papers.- Some Meeting Points of Grobner Bases and Combinatorics.- A Construction of Isomorphism Classes of Oriented Matroids.- Algorithmic Approach to Non-symmetric 3-class Association Schemes.- Sets of Type (d 1,d 2) in Projective Hjelmslev Planes over Galois Rings.- A Construction of Designs from PSL(2,q) and PGL(2,q), q=1 mod 6, on q+2 Points.- Approaching Some Problems in Finite Geometry Through Algebraic Geometry.- Computer Aided Investigation of Total Graph Coherent Configurations for Two Infinite Families of Classical Strongly Regular Graphs.