Directed Strongly Regular Cayley Graphs over Metacyclic Groups of Order 4<i>n</i>

We construct several new families of directed strongly regular Cayley graphs (DSRCGs) over the metacyclic group <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>M</mi> <mrow> <mn>4</mn> <mi>n</mi>...

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Main Authors:	Tao Cheng, Lihua Feng, Weijun Liu
Format:	Article
Language:	English
Published:	MDPI AG 2019-10-01
Series:	Mathematics
Subjects:	directed strongly regular graph cayley graph metacyclic groups
Online Access:	https://www.mdpi.com/2227-7390/7/11/1011

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spelling	doaj-41e0ed9ac4c74bcfb1c9c16fd91e4a232020-11-25T01:56:34ZengMDPI AGMathematics2227-73902019-10-01711101110.3390/math7111011math7111011Directed Strongly Regular Cayley Graphs over Metacyclic Groups of Order 4<i>n</i>Tao Cheng0Lihua Feng1Weijun Liu2School of Mathematics and Statistics, Shandong Normal University, Jinan 250014, ChinaSchool of Mathematics and Statistics, Central South University New Campus, Changsha 410083, ChinaSchool of Mathematics and Statistics, Central South University New Campus, Changsha 410083, ChinaWe construct several new families of directed strongly regular Cayley graphs (DSRCGs) over the metacyclic group <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>M</mi> <mrow> <mn>4</mn> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow> <mo>〈</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mspace width="3.33333pt"></mspace> <mo>\|</mo> <mspace width="3.33333pt"></mspace> <msup> <mi>a</mi> <mi>n</mi> </msup> <mo>=</mo> <msup> <mi>b</mi> <mn>4</mn> </msup> <mo>=</mo> <mn>1</mn> <mo>,</mo> <msup> <mi>b</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>a</mi> <mi>b</mi> <mo>=</mo> <msup> <mi>a</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>〉</mo> </mrow> </mrow> </semantics> </math> </inline-formula>, some of which generalize those earlier constructions. For a prime <i>p</i> and a positive integer <inline-formula> <math display="inline"> <semantics> <mrow> <mi>α</mi> <mo>></mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula>, for some cases, we characterize the DSRCGs over <inline-formula> <math display="inline"> <semantics> <msub> <mi>M</mi> <mrow> <mn>4</mn> <msup> <mi>p</mi> <mi>α</mi> </msup> </mrow> </msub> </semantics> </math> </inline-formula>.https://www.mdpi.com/2227-7390/7/11/1011directed strongly regular graphcayley graphmetacyclic groups
collection	DOAJ
language	English
format	Article
sources	DOAJ
author	Tao Cheng Lihua Feng Weijun Liu
spellingShingle	Tao Cheng Lihua Feng Weijun Liu Directed Strongly Regular Cayley Graphs over Metacyclic Groups of Order 4<i>n</i> Mathematics directed strongly regular graph cayley graph metacyclic groups
author_facet	Tao Cheng Lihua Feng Weijun Liu
author_sort	Tao Cheng
title	Directed Strongly Regular Cayley Graphs over Metacyclic Groups of Order 4<i>n</i>
title_short	Directed Strongly Regular Cayley Graphs over Metacyclic Groups of Order 4<i>n</i>
title_full	Directed Strongly Regular Cayley Graphs over Metacyclic Groups of Order 4<i>n</i>
title_fullStr	Directed Strongly Regular Cayley Graphs over Metacyclic Groups of Order 4<i>n</i>
title_full_unstemmed	Directed Strongly Regular Cayley Graphs over Metacyclic Groups of Order 4<i>n</i>
title_sort	directed strongly regular cayley graphs over metacyclic groups of order 4<i>n</i>
publisher	MDPI AG
series	Mathematics
issn	2227-7390
publishDate	2019-10-01
description	We construct several new families of directed strongly regular Cayley graphs (DSRCGs) over the metacyclic group <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>M</mi> <mrow> <mn>4</mn> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow> <mo>〈</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mspace width="3.33333pt"></mspace> <mo>\|</mo> <mspace width="3.33333pt"></mspace> <msup> <mi>a</mi> <mi>n</mi> </msup> <mo>=</mo> <msup> <mi>b</mi> <mn>4</mn> </msup> <mo>=</mo> <mn>1</mn> <mo>,</mo> <msup> <mi>b</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>a</mi> <mi>b</mi> <mo>=</mo> <msup> <mi>a</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>〉</mo> </mrow> </mrow> </semantics> </math> </inline-formula>, some of which generalize those earlier constructions. For a prime <i>p</i> and a positive integer <inline-formula> <math display="inline"> <semantics> <mrow> <mi>α</mi> <mo>></mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula>, for some cases, we characterize the DSRCGs over <inline-formula> <math display="inline"> <semantics> <msub> <mi>M</mi> <mrow> <mn>4</mn> <msup> <mi>p</mi> <mi>α</mi> </msup> </mrow> </msub> </semantics> </math> </inline-formula>.
topic	directed strongly regular graph cayley graph metacyclic groups
url	https://www.mdpi.com/2227-7390/7/11/1011
work_keys_str_mv	AT taocheng directedstronglyregularcayleygraphsovermetacyclicgroupsoforder4ini AT lihuafeng directedstronglyregularcayleygraphsovermetacyclicgroupsoforder4ini AT weijunliu directedstronglyregularcayleygraphsovermetacyclicgroupsoforder4ini
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Directed Strongly Regular Cayley Graphs over Metacyclic Groups of Order 4<i>n</i>

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