The quadratic unitary Cayley graph <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11128_2025_4877_Article_IEq1.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="28" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {G}_{{\mathbb Z}_n}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">G</mi> <msub> <mi mathvariant="double-struck">Z</mi> <mi>n</mi> </msub> </msub> </math></EquationSource> </InlineEquation> has vertex set <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11128_2025_4877_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="167" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {Z}_n: =\{0,1, \ldots ,n-1\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="double-struck">Z</mi> <mi>n</mi> </msub> <mo>:</mo> <mo>=</mo> <mrow> <mo stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">}</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, where two vertices u and v are adjacent if and only if <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11128_2025_4877_Article_IEq3.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$u - v$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>u</mi> <mo>-</mo> <mi>v</mi> </mrow> </math></EquationSource> </InlineEquation> or <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11128_2025_4877_Article_IEq4.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$v-u$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>v</mi> <mo>-</mo> <mi>u</mi> </mrow> </math></EquationSource> </InlineEquation> is a square of some units in <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11128_2025_4877_Article_IEq5.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {Z}_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">Z</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation>. This paper explores the periodicity and perfect state transfer of Grover walks on quadratic unitary Cayley graphs. We determine all periodic quadratic unitary Cayley graphs. From our results, it follows that there are infinitely many integral as well as nonintegral graphs that are periodic. Additionally, we determine the values of n for which the quadratic unitary Cayley graph <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11128_2025_4877_Article_IEq1.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="28" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {G}_{{\mathbb Z}_n}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">G</mi> <msub> <mi mathvariant="double-struck">Z</mi> <mi>n</mi> </msub> </msub> </math></EquationSource> </InlineEquation> exhibits perfect state transfer.

Periodicity and perfect state transfer of Grover walks on quadratic unitary Cayley graphs

期刊论文
发表日期：

2025年8月11日

查看全文

Koushik Bhakta,
Bikash Bhattacharjya

摘要

The quadratic unitary Cayley graph $\mathcal {G}_{{\mathbb Z}_n}$ $G_{Z_{n}}$ has vertex set $\mathbb {Z}_n: =\{0,1, \ldots ,n-1\}$ $Z_{n} : = {0, 1, \dots, n - 1}$ , where two vertices u and v are adjacent if and only if $u - v$ $u - v$ or $v-u$ $v - u$ is a square of some units in $\mathbb {Z}_n$ $Z_{n}$ . This paper explores the periodicity and perfect state transfer of Grover walks on quadratic unitary Cayley graphs. We determine all periodic quadratic unitary Cayley graphs. From our results, it follows that there are infinitely many integral as well as nonintegral graphs that are periodic. Additionally, we determine the values of n for which the quadratic unitary Cayley graph $\mathcal {G}_{{\mathbb Z}_n}$ $G_{Z_{n}}$ exhibits perfect state transfer.