Alexander Makhnev

• Professor, Corresp. member RAS
• Researcher at Institute of Mathematics and Mechanics UB Russian Academy of Sciences

Given a set \( X \) of the automorphisms of a graph \( \Gamma \), let \( \operatorname{Fix}(X) \) be the set of all vertices of \( \Gamma \) fixed by each automorphism in \( X \). There are exactly 7 admissible intersection arrays of distance-regular graphs with diameter 3 and degree 44. It was proved early that no graphs exist for five of them. In...

The triangle-free Krein graph Kre\((r)\) is strongly regular with parameters \(((r^{2}+3r)^{2},\)\(r^{3}+3r^{2}+r,0,r^{2}+r)\). The existence of such graphs is known only for \(r=1\) (the complement of the Clebsch graph) and \(r=2\) (the Higman–Sims graph). A.L. Gavrilyuk and A.A. Makhnev proved that the graph Kre\((3)\) does not exist. Later Makhn...

A Shilla graph is a distance-regular graph \( \Gamma \) of diameter 3 and second eigenvalue \( \theta_{1}=a_{3} \). In case \( a=a_{3} \) divides \( k \) put \( b=b(\Gamma)=k/a \). Furthermore, \( a_{1}=a-b \) and \( \Gamma \) has the intersection array \( \{ab,(a+1)(b-1),b_{2};1,c_{2},a(b-1)\} \). Belousov and Makhnev found the admissible intersec...

For a distance-regular graph \(\Gamma\) of diameter \(4\), the graph \(\Delta=\Gamma_{1,2}\) can be strongly regular. In this case, the graph \(\Gamma_{3,4}\) is strongly regular and complementary to \(\Delta\). Finding the intersection array of \(\Gamma\) from the parameters of \(\Gamma_{3,4}\) is an inverse problem. In the present paper, the inve...

Earlier it was proved that some distance-regular graphs of diameter 3 with \(c_2=2\) do not exist. Distance-regular graph \(\varGamma \) with intersection array \(\{17,16,10;1,2,8\}\) has strongly regular graph \(\varGamma _{3}\) (pseudo-geometric graph for the net \(pG_9(17,9)\)). By symmetrizing the arrays of triple intersection numbers, it is pr...

In this paper, we determine the maximum signless Laplacian spectral radius of all graphs which do not contain small books as a subgraph and characterize all extremal graphs. In addition, we give an upper bound of the signless Laplacian spectral radius of all graphs which do not contain intersecting quadrangles as a subgraph.

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...

A \(Q\)-polynomial Shilla graph with \(b = 5\) has intersection arrays \(\{105t,4(21t+1),16(t+1); 1,4 (t+1),84t\}\), \(t\in\{3,4,19\}\). The paper proves that distance-regular graphs with these intersection arrays do not exist. Moreover, feasible intersection arrays of \(Q\)-polynomial Shilla graphs with \(b = 6\) are found.

A distance-regular graph \(\Gamma\) of diameter \(3\) is called a Shilla graph if it has the second eigenvalue \(\theta_{1}=a_{3}\). In this case \(a=a_{3}\) divides \(k\) and we set \(b=b(\Gamma)=k/a\). Koolen and Park obtained the list of intersection arrays for Shilla graphs with \(b=3\). There exist graphs with intersection arrays \(\{12,10,5;1...

Let \(\Gamma\) be a distance-regular graph of diameter 3 with a strongly regular graph \(\Gamma_{3}\). Finding the parameters of \(\Gamma_{3}\) from the intersection array of \(\Gamma\) is a direct problem, and finding the intersection array of \(\Gamma\) from the parameters of \(\Gamma_{3}\) is its inverse. The direct and inverse problems were sol...

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...

Automorphisms of a graph with intersection array {nm − 1, nm− n + m − 1, n − m + 1; 1, 1, nm− n + m − 1} are considered.

If regular graph of degree $k$ and diameter $d$ has $v$ vertices then $$v\le 1+k+k(k-1)+\dots+k(k-1)^{d-1}.$$ Graphs with $v=1+k+k(k-1)+\dots+k(k-1)^{d-1}$ is called Moore graphs. Damerell proved that Moore graph of degree $k\ge 3$ has diameter 2. In this case $v=k^2+1$, graph is strongly regular with $\lambda=0$, $\mu=1$ and either degree $k$ is e...

We point out possible automorphisms of a distance-regular graph Γ with intersection array {55, 54, 2; 1, 1, 54} and spectrum 551, 71617,−1110,−81408.

For a distance-regular graph Γ of diameter 3, the graph Γi can be strongly regular for i = 2 or 3. Finding the parameters of Γi from the intersection array of Γ is a direct problem, and finding the intersection array of Γ from the parameters of Γi is the inverse problem. The direct and inverse problems were solved earlier by A. A. Makhnev and M. S....

A distance-regular graph Γ with intersection array {176, 135, 32, 1; 1, 16, 135, 176} is an AT4-graph. Its antipodal quotient \(\overline {\rm{\Gamma }} \) is a strongly regular graph with parameters (672, 176, 40, 48). In both graphs the neighborhoods of vertices are strongly regular with parameters (176, 40, 12, 8). We study the automorphisms of...

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 inv...

Makhnev and Nirova have found intersection arrays of distance-regular graphs with no more than \(4096\) vertices, in which \(\lambda=2\) and \(\mu=1\). They proposed the program of investigation of distance-regular graphs with \(\lambda=2\) and \(\mu=1\). In this paper the automorphisms of a distance-regular graph with intersection array \(\{39,36,...

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...

We complete the classification of edge-symmetric distance-regular coverings of complete graphs with r /∉ {2, k, (k − 1)/μ} for the case of the almost simple action of an automorphism group of a graph on a set of its antipodal classes; here r is the order of an antipodal class.

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: {b2(b−1)2,(b−1)(b2−b+2)2,b(b−1)4;1,b(b−1)4,b(b−1)22} If Γ is a Q-p...

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 nonpri...

Let Γ be a distance regular graph with intersection array {35, 32, 1; 1, 4, 35} and let G = Aut(Γ) act transitively on the set of vertices of the graph Γ. It is shown that G is a {2, 3}-group.

Recently it was shown that a distance-regular graph in which neighbourhoods of vertices are strongly regular with parameters (99,14,1,2) has intersection array {99,84,1;1,14,99}, {99,84,1;1,12,99} or {99,84,30;1,6,54}. In the present paper we find possible automorphisms of a graph with the intersection array {99,84,30;1,6,54}. It is shown, in parti...

J. Koolen posed the problem of studying distance-regular graphs in which neighborhoods of vertices are strongly regular graphs with nonprincipal eigenvalue at most t for a given positive integer t. This problem was solved earlier for t = 3. In the case t = 4, the problem was reduced to studying graphs in which neighborhoods of vertices have paramet...

We consider undirected graphs without loops and multiple edges. Previously, V. P. Burichenko and A. A. Makhnev [1] found intersection arrays of distance-regular locally cyclic graphs with the number of vertices at most 1000. It is shown that the automorphism group of a graph with intersection array {15, 12, 1; 1, 2, 15}, {35, 32, 1; 1, 2, 35}, {39,...

Koolen and Jurisich defined class of AT4-graphs (tight antipodal graph of diameter 4). Among these graphs available graph with intersection array {288; 245; 48; 1; 1; 24; 245; 288} on v = 1 + 288 + 2940 + 576 + 2 = 3807 vertices. Antipodal quotient of this graph is strongly regular graph with parameters (1269; 288; 42; 72). Both these graphs are lo...

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 \(\lambda=\mu\),...

Suppose that a strongly regular graph Γ with parameters (v, k, λ, μ) has eigenvalues k, r, and s. If the graphs Γ and \(\bar \Gamma \) are connected, then the following inequalities, known as Krein’s conditions, hold: (i) (r + 1)(k + r + 2rs) ≤ (k + r)(s + 1)² and (ii) (s + 1)(k + s + 2rs) ≤ (k + s)(r + 1)². We say that Γ is a Krein graph if one of...

Automorphisms of distance-regular graphs are considered. It is proved that any graph with the intersection array {60, 45, 8; 1, 12, 50} is not vertex symmetric, and any graph with the intersection array {49, 36, 8; 1, 6, 42} is not edge symmetric.

Let M be the class of strongly regular graphs for which μ is a nonprincipal eigenvalue. Note that the neighborhood of any vertex of an AT4-graph lies in M. Parameters of graphs from M were described earlier. We find intersection arrays of small AT4-graphs and of strongly regular graphs corresponding to them.

Let Γ be an antipodal graph with intersection array {2r+1, 2r−2, 1; 1, 2, 2r+1}, where 2r(r + 1) ≤ 4096. If 2r + 1 is a prime power, then Mathon’s scheme provides the existence of an arc-transitive graph with this intersection array. Note that 2r + 1 is not a prime power only for r ∈ {7, 17, 19, 22, 25, 27, 31, 32, 37, 38, 42, 43}. We study automor...

In this paper, we investigate antipodal distance-regular graphs of diameter three and valency q(qd-1 - 1)=(q - 1) with arc- transitive automorphism group which induces an almost simple permuta- tion group on the antipodal classes with the socle isomorphic to PSLd(q); where d ≥ 3. We find that such a graph is necessarily bipartite.

A distance-regular graph Γ with intersection array {204, 175, 48, 1; 1, 12, 175, 204} is an AT4-graph, and the antipodal quotient \(\overline \Gamma \) has parameters (800, 204, 28, 60). Automorphisms of these graphs are found. In particular, neither of the two graphs is arc-transitive.

There are some results concerning t-designs in which the number of points in the intersection of two blocks takes less than t values. For example, if t = 2, then the design is symmetric (in such a design, v = b or, equivalently, k = r). In 1974, B. Gross described t-(v, k, l) designs that, for some integer s, 0 < s < t, do not contain two blocks in...

A strongly α-uniform partial line space of order (s, t) is called an α-partial geometry. If α = t+1, then the geometry is a dual 2-design. Locally triangular and locally Grassman graphs correspond to triangular extensions of certain dual 2-designs, and the class of strongly uniform quasi-biplanes coincides with the class of strongly uniform extensi...

Prime divisors of orders of automorphisms and the fixed point subgraphs of automorphisms of prime orders are studied for a hypothetical distance-regular graph with intersection array (45, 42, 1; 1, 6, 45). It is proved that this graph does not vertex-symmetric.

Recently, Makhnev and Nirova found intersection arrays of distance-regular graphs with \(\lambda =2\) and at most 4096 vertices. In the case of primitive graphs of diameter 3 with \(\mu = 1\) there corresponding arrays are \(\{18,15,9;1,1,10\}\), \(\{33,30,8;1,1,30\}\) or \(\{39,36,4;1,1,36\}\). In this work, possible orders and subgraphs of fixed...

The study of graphs with the property specified in the title has earlier been reduced to the case of neighborhoods with parameters (35, 18, 9, 9), (36, 21, 12, 12), (40, 27, 18,18), (50, 28, 15, 16), (56, 45, 36, 36), and (64, 27, 10, 12). It is proved that a completely regular graph in which the neighborhoods of vertices are strongly regular graph...

We call a strongly regular graph with (Formula presented.) and k = 2(m − 2) a Higman graph. In Higman graphs, the parameter µ takes values 4, 6, 7, and 8. We find possible orders of automorphisms of Higman graphs with µ = 6 and study the structure of fixed-point subgraphs of these automorphisms.

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 g...

A. A. Makhnev and D. V. Paduchikh have found intersection arrays of distance-regular graphs, in which neighborhoods of vertices are strongly-regular graphs with second eigenvalue 3. A. A. Makhnev suggested the program to research of automorphisms of these distanceregular graphs. In this paper it is obtained possible orders and subgraphs of fixed po...

Prime divisors of orders of automorphisms and the fixed point subgraphs of automorphisms of prime orders are studied for a hypothetical distance-regular graph with intersection array f75; 72; 1; 1; 12; 75g. It is proved that this graph does not vertex-symmetric.

Prime divisors of orders of automorphisms and the fixed point subgraphs of automorphisms of prime orders are studied for a hypothetical strongly regular graph with parameters (532,156,30,52). Let Γ be a strongly regular graph with parameters (532,156,30,52) and G = Aut(Γ) be a nonsolvable group acting transitively on the vertex set of Γ. Then G = G...

Let H be a subgroup of a finite group G. H is nearly SS-embedded in G if there exists an S-quasinormal subgroup K of G, such that HK is S-quasinormal in G and H ∩ K ≤ H seG , where H seG is the subgroup of H, generated by all those subgroups of H which are S-quasinormally embedded in G. In this paper, the authors investigate the influence of nearly...

A study was conducted to demonstrate extensions of strongly regular graphs with eigenvalue 2. The investigations considered undirected graphs without loops or multiple edges for the demonstration. Makhnev proposed a program for the study of amply regular graphs in which the neighborhood of each vertex was a strongly regular graph with the given par...

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...

We study antipodal distance-regular graphs of diameter 3 such that their automorphism group acts transitively on the set of pairs (a, b), where {a, b} is an edge of the graph. Since the automorphism group of such graphs acts 2-transitively on the set of antipodal classes, the classification of 2-transitive permutation groups can be used. We classif...

Let Γ be a connected edge-regular graph with parameters (ν, κ, λ), b1 = κ - λ-1. It is well known that if b1 = 1, then Γ is a polygon or a complete multipartite graph with colour classes of order 2. The classification of graphs with b1 ≤ 4 is available. Even in the case b1=5 the study of graphs offers great difficulties. However, the situation is m...

Let Γ be a distance-regular graph of diameter 3 with c2 = 2 (any two vertices with distance 2 between them have exactly two common neighbors). Then the neighborhood Δ of the vertex w in Γ is a partial line space. In view of the Brouwer–Neumaier result either Δ is the union of isolated (λ + 1)-cliques or the degrees of vertices k ≥ λ(λ + 3)/2, and i...

Chernikov’s brief biography and information on the International Conference “Algebra and Linear Optimization” (Yekaterinburg, May 14–19, 2012) dedicated to his 100th birthday are presented.

We consider graphs in which neighborhoods of vertices are isomorphic to a strongly regular graph with the second eigenvalue equal to 2. Amply regular graphs in which neighborhoods of vertices are isomorphic to the Mathieu graph (the strongly regular graph with parameters (77,16,0,4)) are classified.

Let Γ be an edge-symmetric distance-regular covering of a clique. Then the group G = Aut(Γ) acts twice transitively on the set Σ of antipodal classes. We propose a classification for the graphs based on the description of twice transitive permutation groups. This program is realized for a 1 = c 2. In this article we classify graphs in the case when...

This paper continues the study of edge-symmetric antipodal distance-regular graphs of diameter 3 by following the program proposed in [A. A. Makhnev et al., Dokl. Math. 87, No. 1, 15–19 (2013); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 448, No. 1, 22–26 (2013; Zbl 1273.05055)] as based on a classification of twice transitive permutation g...

In this paper, we classify the distance-regular graphs in which the neighborhood of vertices are isomorphic to a strong regular graph with parameters (162,21,0,3).

Let Γ be an edge-symmetric distance-regular graph with intersection array {k, (r-1) c,1;1, c, k}. Let G=Aut(Г). Then G acts two-transitively on the set Σ of antipodal classes (fibres) of Γ. These graphs are classified in the next cases: 1) r=2 or r=k, 2) k=rc+1, 3) G contains an abelian normal subgroup N, regular on Σ (affine case).