| Summary: | Let <i>G</i> be a strongly connected digraph, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>d</mi><mi>G</mi></msub><mrow><mo stretchy="false">(</mo><msub><mi>v</mi><mi>i</mi></msub><mo>,</mo><msub><mi>v</mi><mi>j</mi></msub><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula> denote the distance from 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> to vertex <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>v</mi><mi>j</mi></msub></semantics></math></inline-formula> and be defined as the length of the shortest directed path from <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> to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>v</mi><mi>j</mi></msub></semantics></math></inline-formula> in <i>G</i>. The sum distance between vertices <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> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>v</mi><mi>j</mi></msub></semantics></math></inline-formula> in <i>G</i> is defined as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>s</mi><msub><mi>d</mi><mi>G</mi></msub><mrow><mo stretchy="false">(</mo><msub><mi>v</mi><mi>i</mi></msub><mo>,</mo><msub><mi>v</mi><mi>j</mi></msub><mo stretchy="false">)</mo></mrow><mo>=</mo><msub><mi>d</mi><mi>G</mi></msub><mrow><mo stretchy="false">(</mo><msub><mi>v</mi><mi>i</mi></msub><mo>,</mo><msub><mi>v</mi><mi>j</mi></msub><mo stretchy="false">)</mo></mrow><mo>+</mo><msub><mi>d</mi><mi>G</mi></msub><mrow><mo stretchy="false">(</mo><msub><mi>v</mi><mi>j</mi></msub><mo>,</mo><msub><mi>v</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula>. The sum distance matrix of <i>G</i> is the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></semantics></math></inline-formula> matrix <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>D</mi><mrow><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><mo>=</mo><msub><mrow><mo stretchy="false">(</mo><mi>s</mi><msub><mi>d</mi><mi>G</mi></msub><mrow><mo stretchy="false">(</mo><msub><mi>v</mi><mi>i</mi></msub><mo>,</mo><msub><mi>v</mi><mi>j</mi></msub><mo stretchy="false">)</mo></mrow><mo stretchy="false">)</mo></mrow><mrow><msub><mi>v</mi><mi>i</mi></msub><mo>,</mo><msub><mi>v</mi><mi>j</mi></msub><mo>∈</mo><mi>V</mi><mrow><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow></mrow></msub></mrow></semantics></math></inline-formula>. For vertex <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>v</mi><mi>i</mi></msub><mo>∈</mo><mi>V</mi><mrow><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula>, the sum transmission of <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>, denoted by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><msub><mi>T</mi><mi>G</mi></msub><mrow><mo stretchy="false">(</mo><msub><mi>v</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula> or <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><msub><mi>T</mi><mi>i</mi></msub></mrow></semantics></math></inline-formula>, is the row sum of the sum distance matrix <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>D</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> corresponding to 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>. Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>T</mi><mrow><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><mo>=</mo><mi>diag</mi><mrow><mo stretchy="false">(</mo><mi>S</mi><msub><mi>T</mi><mn>1</mn></msub><mo>,</mo><mi>S</mi><msub><mi>T</mi><mn>2</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><mi>S</mi><msub><mi>T</mi><mi>n</mi></msub><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula> be the diagonal matrix with the vertex sum transmissions of <i>G</i> in the diagonal and zeroes elsewhere. For any real number <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo>≤</mo><mi>α</mi><mo>≤</mo><mn>1</mn></mrow></semantics></math></inline-formula>, the general sum distance matrix of <i>G</i> is defined as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><msub><mi>D</mi><mi>α</mi></msub><mrow><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><mo>=</mo><mi>α</mi><mi>S</mi><mi>T</mi><mrow><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><mo>+</mo><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>α</mi><mo stretchy="false">)</mo></mrow><mi>S</mi><mi>D</mi><mrow><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><mo>.</mo></mrow></semantics></math></inline-formula> The eigenvalues of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><msub><mi>D</mi><mi>α</mi></msub><mrow><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula> are called the general sum distance eigenvalues of <i>G</i>, the spectral radius of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><msub><mi>D</mi><mi>α</mi></msub><mrow><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula>, i.e., the largest eigenvalue of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><msub><mi>D</mi><mi>α</mi></msub><mrow><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula>, is called the general sum distance spectral radius of <i>G</i>, denoted by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>μ</mi><mi>α</mi></msup><mrow><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula>. In this paper, we first give some spectral properties of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><msub><mi>D</mi><mi>α</mi></msub><mrow><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula>. We also characterize the digraph minimizes the general sum distance spectral radius among all strongly connected <i>r</i>-partite digraphs. Moreover, for digraphs that are not sum transmission regular, we give a lower bound on the difference between the maximum vertex sum transmission and the general sum distance spectral radius.
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