| Summary: | The signless Laplacian reciprocal distance matrix for a simple connected graph <i>G</i> is defined as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi><mi>Q</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mi>diag</mi><mo>(</mo><mi>R</mi><mi>H</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo><mo>+</mo><mi>R</mi><mi>D</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula>. Here, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi><mi>D</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula> is the Harary matrix (also called reciprocal distance matrix) while <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>diag</mi><mo>(</mo><mi>R</mi><mi>H</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo></mrow></semantics></math></inline-formula> represents the diagonal matrix of the total reciprocal distance vertices. In the present work, some upper and lower bounds for the second-largest eigenvalue of the signless Laplacian reciprocal distance matrix of graphs in terms of various graph parameters are investigated. Besides, all graphs attaining these new bounds are characterized. Additionally, it is inferred that among all connected graphs with <i>n</i> vertices, the complete graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>K</mi><mi>n</mi></msub></semantics></math></inline-formula> and the graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>K</mi><mi>n</mi></msub><mo>−</mo><mi>e</mi></mrow></semantics></math></inline-formula> obtained from <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>K</mi><mi>n</mi></msub></semantics></math></inline-formula> by deleting an edge <i>e</i> have the maximum second-largest signless Laplacian reciprocal distance eigenvalue.
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