The singularity of two kinds of tricyclic graphs
Let $ G $ be a finite simple graph and let $ A(G) $ be its adjacency matrix. Then $ G $ is $ singular $ if $ A(G) $ is singular. Suppose $ P_{b_{1}}, P_{b_{2}}, P_{b_{3}} $ are three paths with disjoint vertices, where $ b_i\geq 2 (i = 1, 2, 3) $, and at most one of them is 2. Coalescing together on...
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AIMS Press
2023-02-01
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| author | Haicheng Ma Xiaojie You Shuli Li |
| author_facet | Haicheng Ma Xiaojie You Shuli Li |
| author_sort | Haicheng Ma |
| collection | DOAJ |
| description | Let $ G $ be a finite simple graph and let $ A(G) $ be its adjacency matrix. Then $ G $ is $ singular $ if $ A(G) $ is singular. Suppose $ P_{b_{1}}, P_{b_{2}}, P_{b_{3}} $ are three paths with disjoint vertices, where $ b_i\geq 2 (i = 1, 2, 3) $, and at most one of them is 2. Coalescing together one of the two end vertices of each of the three paths, and coalescing together the other end vertex of each of the three paths, the resulting graph is called the $ \theta $-graph, denoted by $ \theta(b_{1}, b_{2}, b_{3}) $. Let $ \alpha(a, b_{1}, b_{2}, b_{3}, s) $ be the graph obtained by merging one end of the path $ P_{s} $ with one vertex of a cycle $ C_{a} $, and merging the other end of the path $ P_{s} $ with one vertex of $ \theta(b_{1}, b_{2}, b_{3}) $ of degree 3. If $ s = 1 $, denote $ \beta(a, b_{1}, b_{2}, b_{3}) = \alpha(a, b_{1}, b_{2}, b_{3}, 1) $. In this paper, we give the necessity and sufficiency condition for the singularity of $ \alpha(a, b_{1}, b_{2}, b_{3}, s) $ and $ \beta(a, b_{1}, b_{2}, b_{3}) $, and we also prove that the probability that any given $ \alpha(a, b_{1}, b_{2}, b_{3}, s) $ is a singular graph is equal to $ \frac{35}{64} $, the probability that any given $ \beta(a, b_{1}, b_{2}, b_{3}) $ is a singular graph is equal to $ \frac{9}{16} $. From our main results we can conclude that such a $ \alpha(a, b_{1}, b_{2}, b_{3}, s) $ graph ($ \beta(a, b_{1}, b_{2}, b_{3}) $ graph) is singular if $ 4|a $ or three $ b_i (i = 1, 2, 3) $ are all odd numbers or exactly two of the three $ b_i (i = 1, 2, 3) $ are odd numbers and the length of the cycle formed by the two odd paths in $ \alpha(a, b_{1}, b_{2}, b_{3}, s) $ graph ($ \beta(a, b_{1}, b_{2}, b_{3}) $ graph) is a multiple of 4. The theoretical probability of these graphs being singular is more than half. |
| first_indexed | 2024-04-10T06:23:10Z |
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| spelling | doaj.art-bfbfd07c0abb41e9a2a2a78bfe06e26c2023-03-02T01:17:56ZengAIMS PressAIMS Mathematics2473-69882023-02-01848949896310.3934/math.2023448The singularity of two kinds of tricyclic graphsHaicheng Ma0Xiaojie You1Shuli Li21. School of Mathematics and Statistics, Qinghai Nationalities University, Xining 810007, China1. School of Mathematics and Statistics, Qinghai Nationalities University, Xining 810007, China2. School of Mathematics and Computer Science, Quanzhou Normal University, Quanzhou 362000, China 3. Key Laboratory of Intelligent Computing and Information Processing, Fujian Province University, Quanzhou 362000, ChinaLet $ G $ be a finite simple graph and let $ A(G) $ be its adjacency matrix. Then $ G $ is $ singular $ if $ A(G) $ is singular. Suppose $ P_{b_{1}}, P_{b_{2}}, P_{b_{3}} $ are three paths with disjoint vertices, where $ b_i\geq 2 (i = 1, 2, 3) $, and at most one of them is 2. Coalescing together one of the two end vertices of each of the three paths, and coalescing together the other end vertex of each of the three paths, the resulting graph is called the $ \theta $-graph, denoted by $ \theta(b_{1}, b_{2}, b_{3}) $. Let $ \alpha(a, b_{1}, b_{2}, b_{3}, s) $ be the graph obtained by merging one end of the path $ P_{s} $ with one vertex of a cycle $ C_{a} $, and merging the other end of the path $ P_{s} $ with one vertex of $ \theta(b_{1}, b_{2}, b_{3}) $ of degree 3. If $ s = 1 $, denote $ \beta(a, b_{1}, b_{2}, b_{3}) = \alpha(a, b_{1}, b_{2}, b_{3}, 1) $. In this paper, we give the necessity and sufficiency condition for the singularity of $ \alpha(a, b_{1}, b_{2}, b_{3}, s) $ and $ \beta(a, b_{1}, b_{2}, b_{3}) $, and we also prove that the probability that any given $ \alpha(a, b_{1}, b_{2}, b_{3}, s) $ is a singular graph is equal to $ \frac{35}{64} $, the probability that any given $ \beta(a, b_{1}, b_{2}, b_{3}) $ is a singular graph is equal to $ \frac{9}{16} $. From our main results we can conclude that such a $ \alpha(a, b_{1}, b_{2}, b_{3}, s) $ graph ($ \beta(a, b_{1}, b_{2}, b_{3}) $ graph) is singular if $ 4|a $ or three $ b_i (i = 1, 2, 3) $ are all odd numbers or exactly two of the three $ b_i (i = 1, 2, 3) $ are odd numbers and the length of the cycle formed by the two odd paths in $ \alpha(a, b_{1}, b_{2}, b_{3}, s) $ graph ($ \beta(a, b_{1}, b_{2}, b_{3}) $ graph) is a multiple of 4. The theoretical probability of these graphs being singular is more than half.https://www.aimspress.com/article/doi/10.3934/math.2023448?viewType=HTMLadjacency matrixsingular graphsprobabilitynullity |
| spellingShingle | Haicheng Ma Xiaojie You Shuli Li The singularity of two kinds of tricyclic graphs AIMS Mathematics adjacency matrix singular graphs probability nullity |
| title | The singularity of two kinds of tricyclic graphs |
| title_full | The singularity of two kinds of tricyclic graphs |
| title_fullStr | The singularity of two kinds of tricyclic graphs |
| title_full_unstemmed | The singularity of two kinds of tricyclic graphs |
| title_short | The singularity of two kinds of tricyclic graphs |
| title_sort | singularity of two kinds of tricyclic graphs |
| topic | adjacency matrix singular graphs probability nullity |
| url | https://www.aimspress.com/article/doi/10.3934/math.2023448?viewType=HTML |
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