| Summary: | For any graph <i>G</i> of order <i>p</i>, a bijection <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>:</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>→</mo><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi>…</mi><mo>,</mo><mi>p</mi><mo>}</mo></mrow></semantics></math></inline-formula> is called a numbering of <i>G</i>. The strength <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>s</mi><mi>t</mi><msub><mi>r</mi><mi>f</mi></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> of a numbering <i>f</i> of <i>G</i> is defined by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>s</mi><mi>t</mi><msub><mi>r</mi><mi>f</mi></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mo movablelimits="true" form="prefix">max</mo><mrow><mo>{</mo><mi>f</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>+</mo><mi>f</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mspace width="0.277778em"></mspace><mo>|</mo><mspace width="0.277778em"></mspace><mi>u</mi><mi>v</mi><mo>∈</mo><mi>E</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>}</mo></mrow><mo>,</mo></mrow></semantics></math></inline-formula> and the strength <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>s</mi><mi>t</mi><mi>r</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula> of a graph <i>G</i> is <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>s</mi><mi>t</mi><mi>r</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mo movablelimits="true" form="prefix">min</mo><mo>{</mo><mi>s</mi><mi>t</mi><msub><mi>r</mi><mi>f</mi></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mspace width="0.277778em"></mspace><mo>|</mo><mspace width="0.277778em"></mspace><mi>f</mi><mrow><mi>i</mi><mi>s</mi><mi>a</mi><mi>n</mi><mi>u</mi><mi>m</mi><mi>b</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>o</mi><mi>f</mi></mrow><mi>G</mi><mo>}</mo><mo>.</mo></mrow></semantics></math></inline-formula> In this paper, many open problems are solved, and the strengths of new families of graphs are determined.
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