Resolving an Open Problem on the Exponential Arithmetic–Geometric Index of Unicyclic Graphs
Recently, the exponential arithmetic–geometric index (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>E</mi><mi>A</mi><mi>G</mi></mrow></semantics></m...
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2025-04-01
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| author | Kinkar Chandra Das Jayanta Bera |
| author_facet | Kinkar Chandra Das Jayanta Bera |
| author_sort | Kinkar Chandra Das |
| collection | DOAJ |
| description | Recently, the exponential arithmetic–geometric index (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>E</mi><mi>A</mi><mi>G</mi></mrow></semantics></math></inline-formula>) was introduced. The exponential arithmetic–geometric index (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>E</mi><mi>A</mi><mi>G</mi></mrow></semantics></math></inline-formula>) of a graph <i>G</i> is defined as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>E</mi><mi>A</mi><mi>G</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mstyle displaystyle="true"><munder><mo>∑</mo><mrow><msub><mi>v</mi><mi>i</mi></msub><msub><mi>v</mi><mi>j</mi></msub><mo>∈</mo><mi>E</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></munder></mstyle><mspace width="0.166667em"></mspace><msup><mi>e</mi><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><msub><mi>d</mi><mi>i</mi></msub><mo>+</mo><msub><mi>d</mi><mi>j</mi></msub></mrow><mrow><mn>2</mn><msqrt><mrow><msub><mi>d</mi><mi>i</mi></msub><msub><mi>d</mi><mi>j</mi></msub></mrow></msqrt></mrow></mfrac></mstyle></msup></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>d</mi><mi>i</mi></msub></semantics></math></inline-formula> represents the degree of the vertex <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>v</mi><mi>i</mi></msub></semantics></math></inline-formula> in <i>G</i>. The characterization of extreme structures in relation to graph invariants from the class of unicyclic graphs is an important problem in discrete mathematics. Cruz et al., 2022 proposed a unified method for finding extremal unicyclic graphs for exponential degree-based graph invariants. However, in the case of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>E</mi><mi>A</mi><mi>G</mi></mrow></semantics></math></inline-formula>, this method is insufficient for generating the maximal unicyclic graph. Consequently, the same article presented an open problem for the investigation of the maximal unicyclic graph with respect to this invariant. This article completely characterizes the maximal unicyclic graph in relation to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>E</mi><mi>A</mi><mi>G</mi></mrow></semantics></math></inline-formula>. |
| format | Article |
| id | doaj-art-fd5d2af0d4aa44128a795a5e24e63368 |
| institution | DOAJ |
| issn | 2227-7390 |
| language | English |
| publishDate | 2025-04-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj-art-fd5d2af0d4aa44128a795a5e24e633682025-08-20T02:59:11ZengMDPI AGMathematics2227-73902025-04-01139139110.3390/math13091391Resolving an Open Problem on the Exponential Arithmetic–Geometric Index of Unicyclic GraphsKinkar Chandra Das0Jayanta Bera1Department of Mathematics, Sungkyunkwan University, Suwon 16419, Republic of KoreaDepartment of Mathematics, Sungkyunkwan University, Suwon 16419, Republic of KoreaRecently, the exponential arithmetic–geometric index (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>E</mi><mi>A</mi><mi>G</mi></mrow></semantics></math></inline-formula>) was introduced. The exponential arithmetic–geometric index (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>E</mi><mi>A</mi><mi>G</mi></mrow></semantics></math></inline-formula>) of a graph <i>G</i> is defined as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>E</mi><mi>A</mi><mi>G</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mstyle displaystyle="true"><munder><mo>∑</mo><mrow><msub><mi>v</mi><mi>i</mi></msub><msub><mi>v</mi><mi>j</mi></msub><mo>∈</mo><mi>E</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></munder></mstyle><mspace width="0.166667em"></mspace><msup><mi>e</mi><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><msub><mi>d</mi><mi>i</mi></msub><mo>+</mo><msub><mi>d</mi><mi>j</mi></msub></mrow><mrow><mn>2</mn><msqrt><mrow><msub><mi>d</mi><mi>i</mi></msub><msub><mi>d</mi><mi>j</mi></msub></mrow></msqrt></mrow></mfrac></mstyle></msup></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>d</mi><mi>i</mi></msub></semantics></math></inline-formula> represents the degree of the vertex <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>v</mi><mi>i</mi></msub></semantics></math></inline-formula> in <i>G</i>. The characterization of extreme structures in relation to graph invariants from the class of unicyclic graphs is an important problem in discrete mathematics. Cruz et al., 2022 proposed a unified method for finding extremal unicyclic graphs for exponential degree-based graph invariants. However, in the case of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>E</mi><mi>A</mi><mi>G</mi></mrow></semantics></math></inline-formula>, this method is insufficient for generating the maximal unicyclic graph. Consequently, the same article presented an open problem for the investigation of the maximal unicyclic graph with respect to this invariant. This article completely characterizes the maximal unicyclic graph in relation to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>E</mi><mi>A</mi><mi>G</mi></mrow></semantics></math></inline-formula>.https://www.mdpi.com/2227-7390/13/9/1391extremal graphexponential arithmetic–geometric indexunicyclic graph |
| spellingShingle | Kinkar Chandra Das Jayanta Bera Resolving an Open Problem on the Exponential Arithmetic–Geometric Index of Unicyclic Graphs Mathematics extremal graph exponential arithmetic–geometric index unicyclic graph |
| title | Resolving an Open Problem on the Exponential Arithmetic–Geometric Index of Unicyclic Graphs |
| title_full | Resolving an Open Problem on the Exponential Arithmetic–Geometric Index of Unicyclic Graphs |
| title_fullStr | Resolving an Open Problem on the Exponential Arithmetic–Geometric Index of Unicyclic Graphs |
| title_full_unstemmed | Resolving an Open Problem on the Exponential Arithmetic–Geometric Index of Unicyclic Graphs |
| title_short | Resolving an Open Problem on the Exponential Arithmetic–Geometric Index of Unicyclic Graphs |
| title_sort | resolving an open problem on the exponential arithmetic geometric index of unicyclic graphs |
| topic | extremal graph exponential arithmetic–geometric index unicyclic graph |
| url | https://www.mdpi.com/2227-7390/13/9/1391 |
| work_keys_str_mv | AT kinkarchandradas resolvinganopenproblemontheexponentialarithmeticgeometricindexofunicyclicgraphs AT jayantabera resolvinganopenproblemontheexponentialarithmeticgeometricindexofunicyclicgraphs |