The Number of Spanning Trees of General Fan Graphs

Gao Zhinan

School of Mathematical Sciences, Jiangsu University, China.

Lu Xingyan

School of Mathematical Sciences, Jiangsu University, China.

Wei Yuxuan

School of Mathematical Sciences, Jiangsu University, China.

Zhu Feng *

School of Mathematical Sciences, Jiangsu University, China.

*Author to whom correspondence should be addressed.


Abstract

As we all know, the number of spanning trees can be calculated by virtue of the famous Kirchhoff’s matrix-tree theorem. However, it doesn’t work when coming across complex graphs with thousands of edges and vertices or more. Hence, how to obtain accurate solutions of the number of spanning trees of general fan graphs becomes a subject to several studies of many places like computer science, physics and mathematics. In this paper, we focus on calculating different types of graphs generated by adding vertices and edges to the fan graph. Particularly, we define a new graph called "C-graph", which brings a unique angle of view for us to recognize the construction of original graphs. Moreover, we introduce a new iterative relation about the general fan graph, simplifying the calculation. Therefore, we can obtain functions of the number of spanning trees obtained from given fan graphs, which are also suitable for larger and more complex conditions. Finally, we discuss the effect of vertices and edges on the number of spanning trees, finding out that edges have greater impact. Additionally, by using Kirchhoff’s matrix-tree theorem, we verified the rationality of our results.

Keywords: Fan graphs, spanning trees, iterative relations, self-similar graphs, Kirchhoff’s matrix-tree theorem


How to Cite

Zhinan, Gao, Lu Xingyan, Wei Yuxuan, and Zhu Feng. 2023. “The Number of Spanning Trees of General Fan Graphs”. Asian Research Journal of Mathematics 19 (11):79-94. https://doi.org/10.9734/arjom/2023/v19i11755.

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