| Summary: | A <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mn>6</mn><mo>)</mo></mrow></semantics></math></inline-formula>-fullerene <i>F</i> is a 2-connected cubic planar graph whose faces are only <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn></mrow></semantics></math></inline-formula>-length and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>6</mn></mrow></semantics></math></inline-formula>-length. Furthermore, it consists of exactly three <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn></mrow></semantics></math></inline-formula>-length faces by Euler’s formula. The <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mn>6</mn><mo>)</mo></mrow></semantics></math></inline-formula>-fullerene comes from Došlić’s <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>k</mi><mo>,</mo><mn>6</mn><mo>)</mo></mrow></semantics></math></inline-formula>-fullerene, a 2-connected 3-regular plane graph with only <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi></mrow></semantics></math></inline-formula>-length faces and hexagons. Došlić showed that the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>k</mi><mo>,</mo><mn>6</mn><mo>)</mo></mrow></semantics></math></inline-formula>-fullerenes only exist for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo>=</mo><mn>2</mn></mrow></semantics></math></inline-formula>, 3, 4, or 5, and some of the structural properties of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>k</mi><mo>,</mo><mn>6</mn><mo>)</mo></mrow></semantics></math></inline-formula>-fullerene for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo>=</mo><mn>3</mn></mrow></semantics></math></inline-formula>, 4, or 5 were studied. The structural properties, such as connectivity, extendability, resonance, and anti−Kekulé number, are very useful for studying the number of perfect matchings in a graph, and thus for the study of the stability of the molecular graphs. In this paper, we study the properties of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mn>6</mn><mo>)</mo></mrow></semantics></math></inline-formula>-fullerene. We discover that the edge-connectivity of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mn>6</mn><mo>)</mo></mrow></semantics></math></inline-formula>-fullerenes is 2. Every <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mn>6</mn><mo>)</mo></mrow></semantics></math></inline-formula>-fullerene is 1-extendable, but not 2-extendable (<i>F</i> is called <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mtext>-</mtext></mrow></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>e</mi><mi>x</mi><mi>t</mi><mi>e</mi><mi>n</mi><mi>d</mi><mi>a</mi><mi>b</mi><mi>l</mi><mi>e</mi></mrow></semantics></math></inline-formula> (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>|</mo><mi>V</mi><mo>(</mo><mi>F</mi><mo>)</mo><mo>|</mo><mo>≥</mo><mn>2</mn><mi>n</mi><mo>+</mo><mn>2</mn></mrow></semantics></math></inline-formula>) if any matching of <i>n</i> edges is contained in a perfect matching of <i>F</i>). <i>F</i> is said to be <i>k</i>-<i>resonant</i> (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo>≥</mo><mn>1</mn></mrow></semantics></math></inline-formula>) if the deleting of any <i>i</i> (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>k</mi></mrow></semantics></math></inline-formula>) disjoint even faces of <i>F</i> results in a graph with at least one perfect matching. We have that every <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mn>6</mn><mo>)</mo></mrow></semantics></math></inline-formula>-fullerene is 1-resonant. An edge set, <i>S</i>, of <i>F</i> is called an anti−Kekulé set if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><mo>−</mo><mi>S</mi></mrow></semantics></math></inline-formula> is connected and has no perfect matchings, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><mo>−</mo><mi>S</mi></mrow></semantics></math></inline-formula> denotes the subgraph obtained by deleting all edges in <i>S</i> from <i>F</i>. The anti−Kekulé number of <i>F</i>, denoted by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mi>k</mi><mo>(</mo><mi>F</mi><mo>)</mo></mrow></semantics></math></inline-formula>, is the cardinality of a smallest anti−Kekulé set of <i>F</i>. We have that every <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mn>6</mn><mo>)</mo></mrow></semantics></math></inline-formula>-fullerene <i>F</i> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>|</mo><mi>V</mi><mo>(</mo><mi>F</mi><mo>)</mo><mo>|</mo><mo>></mo><mn>6</mn></mrow></semantics></math></inline-formula> has anti−Kekulé number 4. Further we mainly prove that there exists a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mn>6</mn><mo>)</mo></mrow></semantics></math></inline-formula>-fullerene <i>F</i> having <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>f</mi><mi>F</mi></msub></semantics></math></inline-formula> hexagonal faces, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>f</mi><mi>F</mi></msub></semantics></math></inline-formula> is related to the two parameters <i>n</i> and <i>m</i>.
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