| Summary: | This paper initiates the study of the mathematical aspects of the ad-hoc Lanzhou index. If <i>G</i> is a graph with the vertex set <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{</mo><msub><mi>x</mi><mn>1</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>x</mi><mi>n</mi></msub><mo>}</mo></mrow></semantics></math></inline-formula>, then the ad-hoc Lanzhou index of <i>G</i> is defined by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mover accent="true"><mrow><mi>L</mi><mi>z</mi></mrow><mo>˜</mo></mover><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><msub><mi>d</mi><mi>i</mi></msub><msup><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>−</mo><msub><mi>d</mi><mi>i</mi></msub><mo>)</mo></mrow><mn>2</mn></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>x</mi><mi>i</mi></msub></semantics></math></inline-formula>. Several identities for the ad-hoc Lanzhou index, involving some existing topological indices, are established. The problems of finding graphs with the extremum values of the ad-hoc Lanzhou index from the following sets of graphs are also attacked: (i) the set of all connected <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ξ</mi></semantics></math></inline-formula>-cyclic graphs of a fixed order, (ii) the set of all connected molecular <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ξ</mi></semantics></math></inline-formula>-cyclic graphs of a fixed order, (iii) the set of all graphs of a fixed order, and (iv) the set of all connected molecular graphs of a fixed order.
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