关于连通图的距离能量之间的比较。
On comparison between the distance energies of a connected graph.
作者信息
Ganie Hilal A, Rather Bilal Ahmad, Shang Yilun
机构信息
Department of School Education JK Govt. Kashmir, India.
Department of Mathematical Sciences, Samarkand International University of Technology, Samarkand 140100, Uzbekistan.
出版信息
Heliyon. 2024 Nov 13;10(22):e40316. doi: 10.1016/j.heliyon.2024.e40316. eCollection 2024 Nov 30.
Let be a simple connected graph of order having Wiener index . The distance, distance Laplacian and the distance signless Laplacian energies of are respectively defined as where and are respectively the distance, distance Laplacian and the distance signless Laplacian eigenvalues of and is the average transmission degree. In this paper, we will study the relation between , and . We obtain some necessary conditions for the inequalities and to hold. We will show for graphs with one positive distance eigenvalue the inequality always holds. Further, we will show for the complete bipartite graphs the inequality holds. We end this paper by computational results on graphs of order at most 6.
设(G)是一个阶数为(n)且具有维纳指数(W(G))的简单连通图。(G)的距离、距离拉普拉斯和距离无符号拉普拉斯能量分别定义为(E_d(G)=\sum_{i = 1}^{n}|d_i - \frac{2W(G)}{n}|)，(E_{Ld}(G)=\sum_{i = 1}^{n}|q_i - \frac{2W(G)}{n}|)和(E_{Qd}(G)=\sum_{i = 1}^{n}|Q_i - \frac{2W(G)}{n}|)，其中(d_i)、(q_i)和(Q_i)分别是(G)的距离、距离拉普拉斯和距离无符号拉普拉斯特征值，且(\frac{2W(G)}{n})是平均传输度。在本文中，我们将研究(E_d(G))、(E_{Ld}(G))和(E_{Qd}(G))之间的关系。我们得到了不等式(E_d(G) \leq E_{Ld}(G))和(E_d(G) \leq E_{Qd}(G))成立的一些必要条件。我们将证明对于具有一个正距离特征值的图，不等式(E_d(G) \leq E_{Ld}(G))总是成立。此外，我们将证明对于完全二部图，不等式(E_d(G) \leq E_{Qd}(G))成立。我们通过对阶数至多为(6)的图的计算结果来结束本文。
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