| Summary: | If <inline-formula> <math display="inline"> <semantics> <mrow> <mi>G</mi> <mo>=</mo> <mo>(</mo> <mi>V</mi> <mo>(</mo> <mi>G</mi> <mo>)</mo> <mo>,</mo> <mi>E</mi> <mo>(</mo> <mi>G</mi> <mo>)</mo> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> is a simple connected graph with the vertex set <inline-formula> <math display="inline"> <semantics> <mrow> <mi>V</mi> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> and the edge set <inline-formula> <math display="inline"> <semantics> <mrow> <mi>E</mi> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>, <i>S</i> is a subset of <inline-formula> <math display="inline"> <semantics> <mrow> <mi>V</mi> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>, and let <inline-formula> <math display="inline"> <semantics> <mrow> <mi>B</mi> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> be the set of neighbors of <i>S</i> in <inline-formula> <math display="inline"> <semantics> <mrow> <mi>V</mi> <mo>(</mo> <mi>G</mi> <mo>)</mo> <mo>∖</mo> <mi>S</mi> </mrow> </semantics> </math> </inline-formula>. Then, the differential of <i>S</i><inline-formula> <math display="inline"> <semantics> <mrow> <mi>∂</mi> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> is defined as <inline-formula> <math display="inline"> <semantics> <mrow> <mo>|</mo> <mi>B</mi> <mo>(</mo> <mi>S</mi> <mo>)</mo> <mo>|</mo> <mo>−</mo> <mo>|</mo> <mi>S</mi> <mo>|</mo> </mrow> </semantics> </math> </inline-formula>. The differential of <i>G</i>, denoted by <inline-formula> <math display="inline"> <semantics> <mrow> <mi>∂</mi> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>, is the maximum value of <inline-formula> <math display="inline"> <semantics> <mrow> <mi>∂</mi> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> for all subsets <inline-formula> <math display="inline"> <semantics> <mrow> <mi>S</mi> <mo>⊆</mo> <mi>V</mi> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>. The graph operator <inline-formula> <math display="inline"> <semantics> <mrow> <mo form="prefix">Q</mo> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> is defined as the graph that results by subdividing every edge of <i>G</i> once and joining pairs of these new vertices iff their corresponding edges are incident in <i>G</i>. In this paper, we study the relations between <inline-formula> <math display="inline"> <semantics> <mrow> <mi>∂</mi> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <mi>∂</mi> <mo>(</mo> <mrow> <mo form="prefix">Q</mo> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>. Besides, we exhibit some results relating the differential <inline-formula> <math display="inline"> <semantics> <mrow> <mi>∂</mi> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> and well-known graph invariants, such as the domination number, the independence number, and the vertex-cover number.
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