Vladislav V. Kabanov

Krasovskii Institute of Mathematics and Mechanics, Ural Branch, Russian Academy of Sciences · Department of Algebra and Topology

The symplectic graph $Sp(2d, q)$ is the collinearity graph of the symplectic space of dimension $2d$ over the finite field of order $q$. A $k$-regular graph on $v$ vertices is a divisible design graph with parameters $(v,k,\lambda_1,\lambda_2,m,n)$ if its vertex set can be partitioned into $m$ classes of size $n$, such that any two different vertic...

A k-regular graph on v vertices is a divisible design graph with parameters (v,k,λ1,λ2,m,n) if its vertex set can be partitioned into m classes of size n, such that any two different vertices from the same class have λ1 common neighbours, and any two vertices from different classes have λ2 common neighbours whenever it is not complete or edgeless....

Symplectic graph Sp(2d, q) is the collinearity graph of the symplectic space of dimension 2d over a finite field of order q. A k-regular graph on v vertices is a divisible design graph with parameters (v, k, lambda_1, lambda_2 ,m,n) if its vertex set can be partitioned into m classes of size n, such that any two different vertices from the same cla...

A Deza graph G with parameters (n,k,b,a) is a k-regular graph with n vertices such that any two distinct vertices have b or a common neighbours. The children GA and GB of a Deza graph G are defined on the vertex set of G such that every two distinct vertices are adjacent in GA or GB if and only if they have a or b common neighbours, respectively. A...

A k-regular graph on v vertices is a divisible design graph with parameters (v, k, lambda_1 ,lambda_2, m, n) if its vertex set can be partitioned into m classes of size n, such that any two different vertices from the same class have lambda_1 common neighbours, and any two vertices from different classes have lambda_2 common neighbours whenever it...

Let Δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varDelta$$\end{document} be a digraph of diameter 2 with the maximum undirected vertex degree t and the maximum di...

Deza graph $G$ with parameters $(n,k,b,a)$ is a $k$-regular graph with $n$ vertices for which the number of common neighbours of two distinct vertices takes two possible values, $b$ or $a$, where $b$ is no less than $a$. Moreover, $G$ is not the complete graph or the edgeless graph. Obviously, Deza graphs are a generalisation of strongly regular gr...

A k-regular graph on v vertices is called a divisible design graph with parameters (v,k,\lambda_1 ,\lambda_2 ,m,n) when its vertex set can be partitioned into m classes of size n, such that any two distinct vertices from the same class have \lambda_1 common neighbours, and any two vertices from different classes have \lambda_2 common neighbours. In...

A Deza graph G with parameters (n,k,b,a) is a k-regular graph with n vertices such that any two distinct vertices have b or a common neighbours, where b⩾a. The children GA and GB of a Deza graph G are defined on the vertex set of G such that every two distinct vertices are adjacent in GA or GB if and only if they have a or b common neighbours, resp...

We consider the symmetric group SymΩ with Ω={1,…,n} for any integer n⩾2 and a set S={(1i),i∈{2,…,n}}. The Star graph Sn=Cay(SymΩ,S) is the Cayley graph over the symmetric group SymΩ with the generating set S. For n⩾3, the spectrum of the Star Sn is integral such that for each integer 1⩽k⩽n−1, the values ±(n−k) are its eigenvalues; if n⩾4, then 0 is...

A Deza graph $G$ with parameters $(n,k,b,a)$ is a $k$-regular graph with $n$ vertices such that any two distinct vertices have $b$ or $a$ common neighbours. The children $G_A$ and $G_B$ of a Deza graph $G$ are defined on the vertex set of $G$ such that every two distinct vertices are adjacent in $G_A$ or $G_B$ if and only if they have $a$ or $b$ co...

A Deza graph G with parameters (n,k,b,a) is a k-regular graph with n vertices such that any two distinct vertices have b or a common neighbours, where b >= a. The children G_A and G_B of a Deza graph G are defined on the vertex set of G such that every two distinct vertices are adjacent in G_A or G_B if and only if they have a or b common neighbour...

A Deza graph with parameters (v,k,b,a) is a k‐regular graph on v vertices in which the number of common neighbors of two distinct vertices takes one of the following values: b or a, where b≥a. In the previous papers Deza graphs with b=k−1 were characterized. In this paper, we characterize Deza graphs with b=k−2.

The Star graph $S_n$, $n\ge 3$, is the Cayley graph on the symmetric group $Sym_n$ generated by the set of transpositions $\{(12),(13),\ldots,(1n)\}$. In this work we study eigenfunctions of $S_n$ corresponding to the second largest eigenvalue $n-2$. For $n\ge 8$ and $n=3$, we find the minimum cardinality of the support of an eigenfunction of $S_n$...

A Deza graph with parameters (n,k,b,a) is a k-regular graph with n vertices such that any two of its vertices have b or a common neighbours, where b≥a. In this paper we investigate spectra of Deza graphs. In particular, using the eigenvalues of a Deza graph we determine the eigenvalues of its children. Divisible design graphs are significant cases...

The Star graph $S_n$, $n\ge 3$, is the Cayley graph on the symmetric group $Sym_n$ generated by the set of transpositions $\{(12),(13),\ldots,(1n)\}$. In this work we study eigenfunctions of $S_n$ corresponding to the second largest eigenvalue $n-2$. For $n\ge 8$ and $n=3$, we find the minimum cardinality of the support of an eigenfunction of $S_n$...

We present a construction that gives an infinite series of divisible design graphs which are Cayley graphs.

In this paper we apply dual Seidel switching to the Star graphs and to the Odd graphs, which gives two infinite families of integral graphs. In particular, we obtain three new 4-regular integral connected graphs.

A Deza graph with parameters $(v,k,b,a)$ is a $k$-regular graph on $v$ vertices in which the number of common neighbors of two distinct vertices takes two values $a$ or $b$ ($a\leq b$) and both cases exist. In the previous papers Deza graphs with parameters $(v,k,b,a)$ where $k-b = 1$ were characterized. In the paper we characterise Deza graphs wit...

A Deza graph with parameters is a ‐regular graph with vertices, in which any two vertices have or () common neighbours. A Deza graph is strictly Deza if it has diameter , and is not strongly regular. In an earlier paper, the two last authors et al characterised the strictly Deza graphs with and , where is the number of vertices with common neighbou...

In this paper we present a family of maximal cliques of size [Formula presented] or [Formula presented], accordingly as q≡1(4) or q≡3(4), in Paley graphs of order q², where q is an odd prime power. After that we use the new cliques to define a family of eigenfunctions corresponding to both non-principal eigenvalues and having support size q+1, whic...

A Deza graph with parameters $(n,k,b,a)$ is a $k$-regular graph with $n$ vertices in which any two vertices have $a$ or $b$ ($a\leq b$) common neighbours. A Deza graph is strictly Deza if it has diameter $2$, and is not strongly regular. In an earlier paper, the two last authors et el. characterized the strictly Deza graphs with $b=k-1$ and $\beta...

We consider the symmetric group $\mathrm{Sym}_n,\,n\geqslant 2$, generated by the set $S$ of transpositions $(1~i),\,2 \leqslant i \leqslant n$, and the Cayley graph $S_n=Cay(\mathrm{Sym}_n,S)$ called the Star graph. For any positive integers $n\geqslant 3$ and $m$ with $n > 2m$, we present a family of $PI$-eigenfunctions of $S_n$ with eigenvalue $...

A nonempty $k$-regular graph $\Gamma$ on $n$ vertices is called a Deza graph if there exist constants $b$ and $a$ $(b \geq a)$ such that any pair of distinct vertices of $\Gamma$ has precisely either $b$ or $a$ common neighbours. The quantities $n$, $k$, $b$, and $a$ are called the parameters of $\Gamma$ and are written as the quadruple $(n,k,b,a)$...

A graph Γ is called a Deza graph if it is regular and the number of common neighbors of any two distinct vertices is one of two fixed values. A Deza graph is called a strictly Deza graph if it has diameter 2 and is not strongly regular. In 1992, Gardiner et al. proved that a strongly regular graph that contains a vertex with disconnected second nei...

Граф Γ называется графом Деза, если Γ регулярен и число общих соседей пары произвольных различных вершин принимает одно из двух значений. Точным графом Деза называется граф Деза диаметра 2, не являющийся сильно регулярным графом. В 1992 г. Гарднер (Gardiner), Годсил (Godsil), Хенсел (Hensel) и Ройл (Royle) показали, что сильно регулярный граф, соде...

For a given graph G, its line graph L(G) is defined as the graph with vertex set equal to the edge set of G in which two vertices are adjacent if and only if the corresponding edges of G have exactly one common vertex. A k-regular graph of diameter 2 on υ vertices is called a strictly Deza graph with parameters (υ, k, b, a) if it is not strongly re...

We find the vertex connectivity number of Deza graphs obtained from strongly regular graphs with the use of an automorphism of order 2.

We consider some graphs arising from finite transitive permutation groups.

Let M be a class of strongly regular graphs for which mu is a nonprincipal eigenvalue. Note that the neighborhood of any vertex of an AT4-graph lies in M. We describe parameters of graphs from M and find intersection arrays of AT4-graphs in which neighborhoods of vertices lie in chosen subclasses from M. In particular, an AT4-graph in which neighbo...

A classification is given of connected graphs without 3-claws containing a 3-coclique under some numerical restrictions on the number of vertices in their µ-subgraphs.

It is known that, if the minimal eigenvalue of a graph is −2, then the graph satisfies Hoffman’s condition: for any generated complete bipartite subgraph K 1,3 (a 3-claw) with parts {p} and {q 1, q 2, q 3}, any vertex distinct from p and adjacent to the vertices q 1 and q 2 is adjacent to p but not adjacent to q 3. We prove the converse statement f...

It is known that if the minimal eigenvalue of a graph is −2, then the graph satisfies Hoffman’s condition; i.e., for any generated complete bipartite subgraph K 1,3 with parts {p} and {q 1, q 2, q 3}, any vertex distinct from p and adjacent to two vertices from the second part is not adjacent to the third vertex and is adjacent to p. We prove the c...

Two theorems are proved in this paper. Theorem 1 describes the connected μ-regular graphs without 3-claws. Necessary and sufficient conditions for a connected amply regular graph with μ > 1 to be separated are obtained in Theorem 2. A graph Γ is said to be separated if for any vertex a in Γ the subgraph Γ2(a) contains vertices b and c at a distance...

A subgroup H of a group G is called s-semipermutable in G if H is permutable with every Sylow p-subgroup of G with (p, |H|) = 1. In this paper, we use s-semipermutable subgroups to determine the structure of finite groups. Some of the previous results are generalized.

Some classes of distance-regular graphs (in particular, the graphs of alternating forms and of quadratic forms over the two-element field) in terms of fairly general regularity conditions and some forbidden subgraphs are characterized. Sveral theorems were presented in order to characterize the graphs. Unoriented graphs without loops and multiple e...

The inflation G(I) of a graph G is the line graph of the subdivision of G. If G is a complete graph the equality ir(G(I)) = gamma(G(I)) was proved by Favaron in 1998. We conjectured that the equality holds when G is any graph of radius 1. But it turned out that it is not true. Moreover, we proved that for the class of radius 1 graphs there does not...

The crown is the complete multipartite graph K 1,1,3. Terwilliger graphs without crowns and graphs without 3-cocliques with regular -subgraphs of given positive degree are studied. As a corollary, the local structure of graphs in which the neighborhoods of vertices are regular Terwilliger graphs of diameter 2 and some of these neighborhoods contain...

Anm-crown is the complete tripartite graphK 1, 1,m with parts of order 1, 1,m, and anm-claw is the complete bipartite graphK 1,m with parts of order 1,m, wherem ≥ 3. A vertexa of a graph Γ is calledweakly reduced iff the subgraph {x є Γ ‖a ⊥ =x ⊥} consists of one vertex. A graph Γ is calledweakly reduced iff all its vertices are weakly reduced. In...

We describe coedge regular graphs such that antineighborhoods of their vertices are coedge regular graphs with the same value of the parameterμ. As a consequence of the main theorem, we obtain a classification of coedge regular graphs without 3-stars.

It is proved that the sectional two-rank of a finite group G having no subgroup of index two is at most four if a Sylow two-subgroup of the centralizer of some involution of G is of order 16. This implies the following assertion: If G is a finite simple group whose order is divisible by 25 and the order of the centralizer of some involution of G is...

Let G be a finite simple group with Sylow 2-subgroup T. If there is an extra-special sub-group of index 2 in T, then G is isomorphic to one of the following groups: M11, U3 (q),\mathop {M_{11,} }\limits_{U_3 (q),} for an appropriate odd q.

Without Abstract

Without Abstract