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The Resistance-Harary index of a connected graph <i>G</i> is defined as <inline-formula> <math display="inline"> <semantics> <mrow> <mi>R</mi> <mi>H</mi> <mrow> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> <mo>=</mo> <mstyle displaystyle="true"> <munder> <mo>&#8721;</mo> <mrow> <mrow> <mo>{</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo>}</mo> </mrow> <mo>&sube;</mo> <mi>V</mi> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </munder> </mstyle> <mfrac> <mn>1</mn> <mrow> <mi>r</mi> <mo>(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo>)</mo> </mrow> </mfrac> </mrow> </semantics> </math> </inline-formula>, where <inline-formula> <math display="inline"> <semantics> <mrow> <mi>r</mi> <mo>(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> is the resistance distance between vertices <i>u</i> and <i>v</i> in <i>G</i>. A graph <i>G</i> is called a unicyclic graph if it contains exactly one cycle and a fully loaded unicyclic graph is a unicyclic graph that no vertex with degree less than three in its unique cycle. Let <inline-formula> <math display="inline"> <semantics> <mrow> <mi mathvariant="script">U</mi> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <mi mathvariant="fraktur">U</mi> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> be the set of unicyclic graphs and fully loaded unicyclic graphs of order <i>n</i>, respectively. In this paper, we determine the graphs of <inline-formula> <math display="inline"> <semantics> <mrow> <mi mathvariant="script">U</mi> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> with second-largest Resistance-Harary index and determine the graphs of <inline-formula> <math display="inline"> <semantics> <mrow> <mi mathvariant="fraktur">U</mi> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> with largest Resistance-Harary index.