Pseudogeometric Graph Research Articles

Non-existence of quasi-symmetric designs having certain pseudo-geometric block graphs

The block graph of a quasi-symmetric design is strongly regular. It is a challenging problem to decide which strongly regular graphs are block graphs of quasi-symmetric designs. We rule out the possibility of quasi-symmetric designs with non-zero intersection numbers whose block graphs have the parameters as certain pseudo-geometric graphs.

Automorphisms of a Strongly Regular Graph with Parameters (1305, 440, 115, 165)

A graph Γ is called t-isoregular if, for any i ≤ t and any i-vertex subset S, the number Γ(S) depends only on the isomorphism class of the subgraph induced by S. A graph Γ on v vertices is called absolutely isoregular if it is (v — 1)-isoregular. It is known that each 5-isoregular graph is absolutely isoregular, and such graphs have been fully described. Each precisely 4-isoregular graph is either a pseudogeometric graph for pGr(2r, 2r3 + 3r2 — 1) or its complement. A pseudogeometric graph for pGr(2r, 2r3 + 3r2 — 1) is denoted by Izo(r). Graphs Izo(r) do not exist for an infinite set of values of r (r = 3, 4, 6, 10,…). The existence of Izo(5) is unknown. In this work, we find possible automorphisms for the neighborhood of an edge from Izo(5).

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 {125, 96, 1; 1, 48, 125}

J. Koolen posed the problem of studying distance-regular graphs in which the neighborhoods of vertices are strongly regular graphs whose second eigenvalue is at most t for a given positive integer t. This problem is reduced to the description of distance-regular graphs in which the neighborhoods of vertices are strongly regular graphs with a nonprincipal eigenvalue t for t = 1, 2,.... In the paper “Distance regular graphs in which local subgraphs are strongly regular graphs with the second eigenvalue at most 3”, Makhnev and Paduchikh found intersection arrays of distance-regular graphs in which the neighborhoods of vertices are strongly regular graphs with second eigenvalue t, where 2 < t ≤ 3. Graphs with intersection arrays {125, 96, 1; 1, 48, 125}, {176, 150, 1; 1, 25, 176}, and {256, 204, 1; 1, 51, 256} remained unexplored. In this paper, possible orders and fixed-point subgraphs of automorphisms are found for a distance-regular graph with intersection array {125, 96, 1; 1, 48, 125}. It is proved that the neighborhoods of the vertices of this graph are pseudogeometric graphs for GQ(4, 6). Composition factors of the automorphism group for a distance-regular graph with intersection array {125, 96, 1; 1, 48, 125} are determined.

On the Local Structure of Mathon Distance-Regular Graphs

We study the structure of local subgraphs of distance-regular Mathon graphs of even valency. We describe some infinite series of locally Δ-graphs of this family, where Δ is a strongly regular graph that is the union of affine polar graphs of type “–,” a pseudogeometric graph for p G l (s, l), or a graph of rank 3 realizable by means of the van Lint–Schrijver scheme. We show that some Mathon graphs are characterizable by their intersection arrays in the class of vertex-transitive graphs.

Nonexistence of certain pseudogeometric graphs

Izo(r) is the pseudogeometric graph for pGr(s,t) which satisfies the Krein equality and of which the local graph also satisfies the Krein equality. We prove that Izo(r) exists if and only if a spherical tight 4-design on S(2r+1)2−4 exists. Nonexistence of infinitely many spherical tight 4-designs is well known. It gives the nonexistence of Izo(r) for infinitely many r. Especially, Izo(4) does not exist, which was the first open case.

On extensions of exceptional strongly regular graphs with eigenvalue 3

The study of distance-regular graphs in which neighborhoods of vertices are strongly regular graphs with eigenvalue 3 was initiated by Makhnev. In particular, he reduced these graphs to graphs in which neighborhoods of vertices are exceptional graphs or pseudogeometric graphs for pG s−3(s, t). Makhnev and Paduchikh found parameters of exceptional graphs (see the proposition). In the present paper, we study amply regular graphs in which neighborhoods of vertices are exceptional strongly regular graphs with nonprincipal eigenvalue 3 and parameters from conditions 3–6 of the Proposition.

Graphs of diameter at most 3 whose local subgraphs are pseudogeometric graphs for pG s − 3(s, t)

Graphs of diameter at most 3 whose local subgraphs are pseudogeometric graphs for pG s − 3(s, t)

Exceptional strongly regular graphs with eigenvalue 3

A strongly regular graph Γ with eigenvalue m − 1 is called exceptional if it does not belong to the following list: (1) the union of isolated m-cliques, (2) a pseudogeometric graph for pGt(t+m−1, t), (3) the complement of a pseudogeometric graph for pGm(s,m−1), (4) a graph in the half case with parameters (4µ + 1, 2µ, µ − 1, µ), \(\sqrt {4\mu + 1} = m - 1\). We find parameters of exceptional strongly regular graphs with nonprincipal eigenvalue 3.

On extensions of exceptional strongly regular graphs with eigenvalue 3

The study of distance-regular graphs in which neighborhoods of vertices are strongly regular graphs with eigenvalue 3 was initiated by Makhnev. In particular, he reduced these graphs to graphs in which neighborhoods of vertices are exceptional graphs or pseudogeometric graphs for pG s−3(s, t). Makhnev and Paduchikh found parameters of exceptional graphs (see the proposition). In the present paper, we study amply regular graphs in which neighborhoods of vertices are exceptional strongly regular graphs with nonprincipal eigenvalue 3 and parameters from conditions 3–6 of the Proposition.

Amply regular graphs whose local subgraphs are pseudogeometric graphs for pG s − 3(s, t)

Amply regular graphs whose local subgraphs are pseudogeometric graphs for pG s − 3(s, t)

Exceptional strongly regular graphs with eigenvalue 3

A strongly regular graph Γ with eigenvalue m − 1 is called exceptional if it does not belong to the following list: (1) the union of isolated m-cliques, (2) a pseudogeometric graph for pGt(t+m−1, t), (3) the complement of a pseudogeometric graph for pGm(s,m−1), (4) a graph in the half case with parameters (4µ + 1, 2µ, µ − 1, µ), \(\sqrt {4\mu + 1} = m - 1\). We find parameters of exceptional strongly regular graphs with nonprincipal eigenvalue 3.

Graphs in which the neighborhoods of vertices are pseudogeometric graphs for GQ(3, 5)

Graphs in which the neighborhoods of vertices are pseudogeometric graphs for GQ(3, 5)

Graphs in which the neighborhoods of vertices are pseudogeometric graphs for GQ(3, 3)

Graphs in which the neighborhoods of vertices are pseudogeometric graphs for GQ(3, 3)

On graphs in which the neighborhoods of vertices are pseudogeometric graphs for pG s − 2(s, t)

On graphs in which the neighborhoods of vertices are pseudogeometric graphs for pG s − 2(s, t)

Ovoids and Bipartite Subgraphs in Generalized Quadrangles

A point-line incidence system is called an α-partial geometry of order (s,t) if each line contains s + 1 points, each point lies on t + 1 lines, and for any point a not lying on a line L, there exist precisely α lines passing through a and intersecting L (the notation is pGα(s,t)). If α = 1, then such a geometry is called a generalized quadrangle and denoted by GQ(s,t). It is established that if a pseudogeometric graph for a generalized quadrangle GQ(s,s2 − s) contains more than two ovoids, then s = 2. It is proved that the point graph of a generalized quadrangle GQ(4,t) contains no K4,6-subgraphs. Finally, it is shown that if some μ-subgraph of a pseudogeometric graph for a generalized quadrangle GQ(4,t) contains a triangle, then t ≤ 6.

On Strongly Regular Graphs with k=2μ and Their Extensions

We obtain a convenient expression for the parameters of a strongly regular graph with k=2μ in terms of the nonprincipal eigenvalues x and −y. It turns out in particular that such graphs are pseudogeometric for pGx(2x,y−1). We prove that a strongly regular graph with parameters (35,16,6,8) is a quotient of the Johnson graph \(\bar J\)(8,4). We also find the parameters of strongly regular graphs in which the neighborhoods of vertices are pseudogeometric graphs for pGx(2x,t),x≤3. In consequence, we establish that a connected graph in which the neighborhoods of vertices are pseudogeometric graphs for pG3(6,2) is isomorphic to the Taylor graph on 72 vertices or to the alternating form graph Alt(4,2) with parameters (64,35,18,20).

On the Nonexistence of Strongly Regular Graphs with Parameters (486, 165, 36, 66)

We prove that a strongly regular graph with parameters (486, 165, 36, 66) does not exist. Since the parameters indicated are parameters of a pseudogeometric graph for pG2(5, 32), we conclude that the partial geometries pG2(5, 32) and pG2(32, 5) do not exist. Finally, a neighborhood of an arbitrary vertex of a pseudogeometric graph for pG3(6, 80) is a pseudogeometric graph for pG2(5, 32) and, therefore, a pseudogeometric graph for the partial geometry pG3(6, 80) [i.e., a strongly regular graph with parameters (1127, 486, 165, 243)] does not exist.

The Pseudo-geometric Graphs for Generalized Quadrangles of Order (3, t)

The values t= 1, 3, 5, 6, 9 satisfy the standard necessary conditions for existence of a generalized quadrangle of order (3, t). This gives the following possible parameter sets for strongly regular graphs that are pseudo-geometric for such a generalized quadrangle: (v, k,λ , μ) = (16, 6, 2, 2), (40, 12, 2, 4), (64, 18, 2, 6), (76, 21, 2, 7) and (112, 30, 2, 10). It is well known that there are two graphs with the first parameter set and that there is just one graph with the last set of parameters. Recently, the second author has shown that there are precisely 28 strongly regular graphs with the second parameter set. Non-existence of a strongly regular graph with the fourth set of parameters was proved by the first author. Here we complete the classification by announcing that there are exactly 167 non-isomorphic strongly regular graphs with parameters (64, 18, 2, 6).

On pseudogeometric graphs of the partial geometries

An incidence system consisting of points and lines is called an -partial geometry of order if each line contains points, each point lies on lines (the lines intersect in at most one point), and for any point a not lying on a line there are exactly lines passing through and intersecting (this geometry is denoted by ). The point graph of the partial geometry is strongly regular with parameters: , , and . A graph with the indicated parameters is called a pseudogeometric graph of the corresponding geometry. It is proved that a pseudogeometric graph of a partial geometry in which the -subgraphs are regular graphs without triangles is the triangular graph , the quotient of the Johnson graph , or the McLaughlin graph.