Extra Edge Connectivity and Extremal Problems in Education Networks
Extra edge connectivity and diagnosability have been employed to investigate the fault tolerance properties of network structures. The <i>p</i>-extra edge connectivity <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><se...
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2022-09-01
|
| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/10/19/3475 |
| _version_ | 1827654086102614016 |
|---|---|
| author | Hongfang Liu Jinxia Liang Kinkar Chandra Das |
| author_facet | Hongfang Liu Jinxia Liang Kinkar Chandra Das |
| author_sort | Hongfang Liu |
| collection | DOAJ |
| description | Extra edge connectivity and diagnosability have been employed to investigate the fault tolerance properties of network structures. The <i>p</i>-extra edge connectivity <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>λ</mi><mi>p</mi></msub><mrow><mo>(</mo><mo>Γ</mo><mo>)</mo></mrow></mrow></semantics></math></inline-formula> of a graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Γ</mo></semantics></math></inline-formula> was introduced by Fàbrega and Fiol in 1996. In this paper, we find the exact values of <i>p</i>-extra edge connectivity of some special graphs. Moreover, we give some upper and lower bounds for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>λ</mi><mi>p</mi></msub><mrow><mo>(</mo><mo>Γ</mo><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, and graphs with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>λ</mi><mi>p</mi></msub><mrow><mo>(</mo><mo>Γ</mo><mo>)</mo></mrow><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mfenced separators="" open="⌈" close="⌉"><mfrac><mi>n</mi><mn>2</mn></mfrac></mfenced><mfenced separators="" open="⌊" close="⌋"><mfrac><mi>n</mi><mn>2</mn></mfrac></mfenced><mo>−</mo><mn>1</mn><mo>,</mo><mfenced separators="" open="⌈" close="⌉"><mfrac><mi>n</mi><mn>2</mn></mfrac></mfenced><mfenced separators="" open="⌊" close="⌋"><mfrac><mi>n</mi><mn>2</mn></mfrac></mfenced></mrow></semantics></math></inline-formula> are characterized. Finally, we obtain the three extremal results for the <i>p</i>-extra edge connectivity. |
| first_indexed | 2024-03-09T21:28:37Z |
| format | Article |
| id | doaj.art-72c1138431dd4d4484e48af58bf7212d |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T21:28:37Z |
| publishDate | 2022-09-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-72c1138431dd4d4484e48af58bf7212d2023-11-23T21:02:08ZengMDPI AGMathematics2227-73902022-09-011019347510.3390/math10193475Extra Edge Connectivity and Extremal Problems in Education NetworksHongfang Liu0Jinxia Liang1Kinkar Chandra Das2School of Education, Shaanxi Normal University, Xi’an 710062, ChinaSchool of Mathematics and Statistics, Qinghai Normal University, Xining 810008, ChinaDepartment of Mathematics, Sungkyunkwan University, Suwon 16419, KoreaExtra edge connectivity and diagnosability have been employed to investigate the fault tolerance properties of network structures. The <i>p</i>-extra edge connectivity <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>λ</mi><mi>p</mi></msub><mrow><mo>(</mo><mo>Γ</mo><mo>)</mo></mrow></mrow></semantics></math></inline-formula> of a graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Γ</mo></semantics></math></inline-formula> was introduced by Fàbrega and Fiol in 1996. In this paper, we find the exact values of <i>p</i>-extra edge connectivity of some special graphs. Moreover, we give some upper and lower bounds for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>λ</mi><mi>p</mi></msub><mrow><mo>(</mo><mo>Γ</mo><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, and graphs with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>λ</mi><mi>p</mi></msub><mrow><mo>(</mo><mo>Γ</mo><mo>)</mo></mrow><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mfenced separators="" open="⌈" close="⌉"><mfrac><mi>n</mi><mn>2</mn></mfrac></mfenced><mfenced separators="" open="⌊" close="⌋"><mfrac><mi>n</mi><mn>2</mn></mfrac></mfenced><mo>−</mo><mn>1</mn><mo>,</mo><mfenced separators="" open="⌈" close="⌉"><mfrac><mi>n</mi><mn>2</mn></mfrac></mfenced><mfenced separators="" open="⌊" close="⌋"><mfrac><mi>n</mi><mn>2</mn></mfrac></mfenced></mrow></semantics></math></inline-formula> are characterized. Finally, we obtain the three extremal results for the <i>p</i>-extra edge connectivity.https://www.mdpi.com/2227-7390/10/19/3475connectivityp-extra edge connectivitydiametereducation network |
| spellingShingle | Hongfang Liu Jinxia Liang Kinkar Chandra Das Extra Edge Connectivity and Extremal Problems in Education Networks Mathematics connectivity p-extra edge connectivity diameter education network |
| title | Extra Edge Connectivity and Extremal Problems in Education Networks |
| title_full | Extra Edge Connectivity and Extremal Problems in Education Networks |
| title_fullStr | Extra Edge Connectivity and Extremal Problems in Education Networks |
| title_full_unstemmed | Extra Edge Connectivity and Extremal Problems in Education Networks |
| title_short | Extra Edge Connectivity and Extremal Problems in Education Networks |
| title_sort | extra edge connectivity and extremal problems in education networks |
| topic | connectivity p-extra edge connectivity diameter education network |
| url | https://www.mdpi.com/2227-7390/10/19/3475 |
| work_keys_str_mv | AT hongfangliu extraedgeconnectivityandextremalproblemsineducationnetworks AT jinxialiang extraedgeconnectivityandextremalproblemsineducationnetworks AT kinkarchandradas extraedgeconnectivityandextremalproblemsineducationnetworks |