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A singular graph <i>G</i>, defined when its adjacency matrix is singular, has important applications in mathematics, natural sciences and engineering. The chemical importance of singular graphs lies in the fact that if the molecular graph is singular, the nullity (the number of the zero eigenvalue) is greater than 0, then the corresponding chemical compound is highly reactive or unstable. By this reasoning, chemists have a great interest in this problem. Thus, the problem of characterization singular graphs was proposed and raised extensive studies on this challenging problem thereafter. The graph obtained by conglutinating the starting vertices of three paths <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><msub><mi>s</mi><mn>1</mn></msub></msub></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><msub><mi>s</mi><mn>2</mn></msub></msub></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><msub><mi>s</mi><mn>3</mn></msub></msub></semantics></math></inline-formula> into a vertex, and three end vertices into a vertex on the cycle <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>C</mi><msub><mi>a</mi><mn>1</mn></msub></msub></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>C</mi><msub><mi>a</mi><mn>2</mn></msub></msub></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>C</mi><msub><mi>a</mi><mn>3</mn></msub></msub></semantics></math></inline-formula>, respectively, is denoted as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>γ</mi><mo>(</mo><msub><mi>a</mi><mn>1</mn></msub><mo>,</mo><msub><mi>a</mi><mn>2</mn></msub><mo>,</mo><msub><mi>a</mi><mn>3</mn></msub><mo>,</mo><msub><mi>s</mi><mn>1</mn></msub><mo>,</mo><msub><mi>s</mi><mn>2</mn></msub><mo>,</mo><msub><mi>s</mi><mn>3</mn></msub><mo>)</mo></mrow></semantics></math></inline-formula>. Note that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>δ</mi><mrow><mo>(</mo><msub><mi>a</mi><mn>1</mn></msub><mo>,</mo><msub><mi>a</mi><mn>2</mn></msub><mo>,</mo><msub><mi>a</mi><mn>3</mn></msub><mo>,</mo><msub><mi>s</mi><mn>1</mn></msub><mo>,</mo><msub><mi>s</mi><mn>2</mn></msub><mo>)</mo></mrow><mo>=</mo><mi>γ</mi><mrow><mo>(</mo><msub><mi>a</mi><mn>1</mn></msub><mo>,</mo><msub><mi>a</mi><mn>2</mn></msub><mo>,</mo><msub><mi>a</mi><mn>3</mn></msub><mo>,</mo><msub><mi>s</mi><mn>1</mn></msub><mo>,</mo><mn>1</mn><mo>,</mo><msub><mi>s</mi><mn>2</mn></msub><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ζ</mi><mrow><mo>(</mo><msub><mi>a</mi><mn>1</mn></msub><mo>,</mo><msub><mi>a</mi><mn>2</mn></msub><mo>,</mo><msub><mi>a</mi><mn>3</mn></msub><mo>,</mo><mi>s</mi><mo>)</mo></mrow><mo>=</mo><mi>γ</mi><mrow><mo>(</mo><msub><mi>a</mi><mn>1</mn></msub><mo>,</mo><msub><mi>a</mi><mn>2</mn></msub><mo>,</mo><msub><mi>a</mi><mn>3</mn></msub><mo>,</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mi>s</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>φ</mi><mrow><mo>(</mo><msub><mi>a</mi><mn>1</mn></msub><mo>,</mo><msub><mi>a</mi><mn>2</mn></msub><mo>,</mo><msub><mi>a</mi><mn>3</mn></msub><mo>)</mo></mrow><mo>=</mo><mi>γ</mi><mrow><mo>(</mo><msub><mi>a</mi><mn>1</mn></msub><mo>,</mo><msub><mi>a</mi><mn>2</mn></msub><mo>,</mo><msub><mi>a</mi><mn>3</mn></msub><mo>,</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. In this paper, we give the necessity and sufficiency that the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>γ</mi><mo>−</mo></mrow></semantics></math></inline-formula>graph, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>δ</mi><mo>−</mo></mrow></semantics></math></inline-formula>graph, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ζ</mi><mo>−</mo></mrow></semantics></math></inline-formula>graph and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>φ</mi><mo>−</mo></mrow></semantics></math></inline-formula>graph are singular and prove that the probability that a randomly given <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>γ</mi><mo>−</mo></mrow></semantics></math></inline-formula>graph, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>δ</mi><mo>−</mo></mrow></semantics></math></inline-formula>graph, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ζ</mi><mo>−</mo></mrow></semantics></math></inline-formula>graph or <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>φ</mi><mo>−</mo></mrow></semantics></math></inline-formula>graph being singular is equal to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mfrac><mn>325</mn><mn>512</mn></mfrac><mo>,</mo><mfrac><mn>165</mn><mn>256</mn></mfrac><mo>,</mo><mfrac><mn>43</mn><mn>64</mn></mfrac></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfrac><mn>21</mn><mn>32</mn></mfrac></semantics></math></inline-formula>, respectively. From our main results, we can conclude that such a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>γ</mi><mo>−</mo></mrow></semantics></math></inline-formula>graph(<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>δ</mi><mo>−</mo></mrow></semantics></math></inline-formula>graph, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ζ</mi><mo>−</mo></mrow></semantics></math></inline-formula>graph, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>φ</mi><mo>−</mo></mrow></semantics></math></inline-formula>graph) is singular if at least one cycle is a multiple of 4 in length, and surprisingly, the theoretical probability of these graphs being singular is more than half. This result promotes the understanding of a singular graph and may be promising to propel the solutions to relevant application problems.