<abstract xmlns="http://www.w3.org/1999/xhtml"><p>Balaban index is defined as <inline-formula><alternatives><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mi>J</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mi>m</mi><mrow><mi>m</mi><mo>−</mo><mi>n</mi><mo>+</mo><mn>2</mn></mrow></mfrac><mi>Σ</mi><mfrac><mn>1</mn><mrow><msqrt><mrow><mi>w</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>⋅</mo><mi>w</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow></mrow></msqrt></mrow></mfrac><mo>,</mo></mrow></math><tex-math>$J\left( G \right)=\frac{m}{m-n+2}\Sigma \frac{1}{\sqrt{w\left( u \right)\cdot w\left( v \right)}},$</tex-math></alternatives></inline-formula> where the sum is taken over all edges of a connected graph <italic>G</italic>, <italic>n</italic> and <italic>m</italic> are the cardinalities of the vertex and the edge set of <italic>G</italic>, respectively, and <italic>w</italic>(<italic>u</italic>) (resp. <italic>w</italic>(<italic>v</italic>)) denotes the sum of distances from <italic>u</italic> (resp. <italic>v</italic>) to all the other vertices of <italic>G</italic>. In 2011, H. Deng found an interesting property that Balaban index is a convex function in double stars. We show that this holds surprisingly to general graphs by proving that attaching leaves at two vertices in a graph yields a new convexity property of Balaban index. We demonstrate this property by finding, for each <italic>n</italic>, seven trees with the maximum value of Balaban index, and we conclude the paper with an interesting conjecture.</p></abstract>

Balaban index is defined as J(G)=mm−n+2Σ1w(u)⋅w(v),$J\left( G \right)=\frac{m}{m-n+2}\Sigma \frac{1}{\sqrt{w\left( u \right)\cdot w\left( v \right)}},$ where the sum is taken over all edges of a connected graph G, n and m are the cardinalities of the vertex and the edge set of G, respectively, and w(u) (resp. w(v)) denotes the sum of distances from u (resp. v) to all the other vertices of G. In 2011, H. Deng found an interesting property that Balaban index is a convex function in double stars. We show that this holds surprisingly to general graphs by proving that attaching leaves at two vertices in a graph yields a new convexity property of Balaban index. We demonstrate this property by finding, for each n, seven trees with the maximum value of Balaban index, and we conclude the paper with an interesting conjecture.

Convexity result and trees with large Balaban index

Applied Mathematics and Nonlinear Sciences

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

{"article-title":"Convexity result and trees with large Balaban index"}

Balaban index is defined as J(G)=mm−n+2Σ1w(u)⋅w(v),$J\left( G \right)=\frac{m}{m-n+2}\Sigma \frac{1}{\sqrt{w\left( u \right)\cdot w\left( v \right)}},$ where...

n	T₁	T₂	T₃	T₄	T₅	T₆	T₇
2-3
4
5			P₅
6						P₆
7				H₂,_n-₃,₂	H_n-₃,₂,₂	H₃,₃,₂	R₇
8	S _n			D₄,₄	H₂,₅,₂
9		D_n-₂,₂				H_n-₃,₂,₂	H₃,_n-₄,₂
10			D _n-₃,₃			D₅,₅	H_n-₃,₂,₂
11				H₂,_n-₃,₂	D _n-₄,₄		D₆,₅
12-17						H _n-₃,₂,₂
18-¥					H_n-₃,₂,₂	D_n-₄,₄	H₃,_n-₄,₂

Convexity result and trees with large Balaban index

Publié en ligne: 15 août 2018

Pages: 433 - 446

Reçu: 02 avr. 2018

Accepté: 15 août 2018

DOI: https://doi.org/10.21042/AMNS.2018.2.00034

© 2018 Martin Knor et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Fig. 1

Fig. 2

Fig. 3

First seven trees with maximal values of Balaban index.