Graph-Theoretic Problems and Their New Applications.

Focusing first on the case HT <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi><mo>=</mo><mn>2</mn></mrow></semantics></math> ht , they determine bounds on the number of edges. Finally, the case of a fixed value HT <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi><mo><=</mo><mi>r</mi><mo>-</mo><mn>1</mn></mrow></semantics></math> ht is also discussed. They determine the graphs with largest HT <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>A</mi><mi> </mi></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math> ht -spectral radius with fixed vertex or edge connectivity. In addition, the corresponding extremal graphs are given and equations satisfying the HT <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>A</mi><mi> </mi></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math> ht -spectral radius are derived. [Extracted from the article]