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Alexander L. Gavrilyuk

Publications and source records attributed to Alexander L. Gavrilyuk.

29 records · Page 2Linked to original sources
arXiv2018

On a characterization of the Grassmann graphs

In 1995, Metsch showed that the Grassmann graph $J_q(n,D)$ of diameter $D\geq 3$ is characterized by its intersection numbers with the following possible exceptions: (-) $n=2D$ or $n=2D+1$, $q\geq 2$; (-) $n=2D+2$ and $q\in \{2,3\}$; (-) $n=2D+3$ and $q=2$. In 2005, Van Dam and Koolen constructed the twisted Grassmann graphs with the same intersection numbers as the Grassmann graphs $J_q(2D+1,D)$, for any prime power $q$ and diameter $D\geq 2$, but they are not isomorphic. We show that the Grassmann graph $J_q(2D,D)$ is characterized by its intersection numbers provided that the diameter $D$ is large enough.

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arXiv2018

A characterization of the graphs of bilinear $(d\times d)$-forms over $\mathbb{F}_2$

The bilinear forms graph denoted here by $Bil_q(e\times d)$ is a graph defined on the set of $(e\times d)$-matrices ($e\geq d$) over $\mathbb{F}_q$ with two matrices being adjacent if and only if the rank of their difference equals $1$. In 1999, K. Metsch showed that the bilinear forms graph $Bil_q(e\times d)$ is characterized by its intersection array if one of the following holds: (-) $q=2$ and $e\geq d+4$, (-) $q\geq 3$ and $e\geq d+3$. Thus, the following cases have been left unsettled: (-) $q=2$ and $e\in \{d,d+1,d+2,d+3\}$, (-) $q\geq 3$ and $e\in \{d,d+1,d+2\}$. In this work, we show that the graph of bilinear $(d\times d)$-forms over the binary field, where $d\geq 3$, is characterized by its intersection array. In doing so, we also classify locally grid graphs whose $μ$-graphs are hexagons and the intersection numbers $b_i,c_i$ are well-defined for all $i=0,1,2$.

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arXiv2016

Biregular graphs with three eigenvalues

We consider nonregular graphs having precisely three distinct eigenvalues. The focus is mainly on the case of graphs having two distinct valencies and our results include constructions of new examples, structure theorems, valency constraints, and a classification of certain special families of such graphs. We also present a new example of a graph with three valencies and three eigenvalues of which there are currently only finitely many known examples.

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arXiv2014

The Terwilliger polynomial of a Q-polynomial distance-regular graph and its application to the pseudo-partition graphs

Let $Γ$ be a $Q$-polynomial distance-regular graph with diameter at least $3$. Terwilliger (1993) implicitly showed that there exists a polynomial, say $T(λ)\in \mathbb{C}[λ]$, of degree $4$ depending only on the intersection numbers of $Γ$ and such that $T(η)\geq 0$ holds for any non-principal eigenvalue $η$ of the local graph $Γ(x)$ for any vertex $x\in V(Γ)$. We call $T(λ)$ the Terwilliger polynomial of $Γ$. In this paper, we give an explicit formula for $T(λ)$ in terms of the intersection numbers of $Γ$ and its dual eigenvalues. We then apply this polynomial to show that all pseudo-partition graphs with diameter at least $3$ are known.

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arXiv2012

A note on Cameron - Liebler line classes in PG(n,4)

A {\it Cameron -- Liebler line class} ${\cal L}$ with parameter $x$ is a set of lines of projective geometry $PG(3,q)$ such that each line of ${\cal L}$ meets exactly $x(q+1)+q^2-1$ lines of ${\cal L}$ and each line that is not from ${\cal L}$ meets exactly $x(q+1)$ lines of ${\cal L}$. In this paper, we obtain a classification of Cameron -- Liebler line classes in PG(3,4) and a classification of their generalization in PG(n,4), $n\geqslant 4$.

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