有限循环群的SGB图的某些拓扑指数和谱性质
Certain topological indices and spectral properties of SGB-graphs of finite cyclic groups
AI总结：
本文研究了有限循环群的SGB图的拓扑指数和谱性质，并验证了相关猜想。
摘要：
设L(G)是群G的所有子群的集合。子群生成二分图B(G)定义在G上，其顶点集是两个集合G×G和L(G)的并集，若两个顶点(a, b)∈G×G和H∈L(G)满足H由a和b生成，则它们相邻。在本文中，我们研究了循环群阶数为pq、p²q和p²q²时的B(G)结构，其中p和q为素数且p≠q。我们还推导了这些图的第一和第二Zagreb指数的表达式，并验证了Hansen-Vukičević猜想。此外，我们还计算了这些图的其他度数相关的拓扑指数的表达式。我们进一步计算了当G为任意阶数为pⁿ、pq、p²q和p²q²的循环群时，B(G)的各种谱及其对应的能量。最后，我们证明B(G)满足E-LE猜想。
英文摘要：
Let $L(G)$ be the set of all subgroups of a group $G$. The subgroup generating bipartite graph $\mathcal{B}(G)$ defined on $G$ is a bipartite graph whose vertex set is the union of two sets $G \times G$ and $L(G)$, and two vertices $(a, b) \in G \times G$ and $H \in L(G)$ are adjacent if $H$ is generated by $a$ and $b$. In this paper, we realize the structures of $\mathcal{B}(G)$ for cyclic groups of order $pq, p^2q$ and $p^2q^2$, where $p$ and $q$ are primes and $p \neq q$. We also deduce expressions for first and second Zagreb indices of these graphs and check the validity of Hansen-Vuki{č}evi{ć} conjecture [Hansen, P. and Vuki{č}evi{ć}, D. Comparing the Zagreb indices, {\em Croatica Chemica Acta}, \textbf{80}(2), 165-168, 2007]. Expressions of certain other degree-based topological indices of these graphs are also computed. We further compute various spectra and their corresponding energies of $\mathcal{B}(G)$ if $G$ is any cyclic group of order $p^n, pq, p^2q$ and $p^2q^2$, where $p$ and $q$ are two distinct primes and $n \geq 1$. We conclude the paper showing that $\mathcal{B}(G)$ satisfies E-LE conjecture [Gutman, I., Abreu, N. M. M., Vinagre, C. T. M., Bonifacioa, A. S. and Radenkovic, S. Relation between energy and Laplacian energy, {\em MATCH Communications in Mathematical and in Computer Chemistry}, \textbf{59}, 343--354, 2008] for these groups.
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