能量最小化化学树与Hosoya指标
Chemical trees minimizing energy and Hosoya index
摘要：
分子图的能量是一个常用的参数，它被定义为图的特征值的绝对值之和。众所周知，能量与匹配多项式相关，因此也通过一定的Coulson积分与Hosoya指数相关。在过去已经确定了在各种附加条件下使能量最小化的树，例如，给定直径的树或完美匹配的树。然而，最小化具有有界最大度的树的能量是一个很自然的问题-显然，最大度为4的情况（所谓的化学树）是最重要的一个。我们将表明，树与给定的最大程度，最小化的能量是相同的，已经显示以前最小化Hosoya指数和最大化的Merrifield-Simmons指数，从而也证明了一个猜想，由于Fischermann等人。最后，我们表明，最小的能量增长与树的大小呈线性关系，明确可计算的增长常数，只依赖于最大程度。
英文摘要：
The energy of a molecular graph is a popular parameter that is defined as the sum of the absolute values of a graph's eigenvalues. It is well known that the energy is related to the matching polynomial and thus also to the Hosoya index via a certain Coulson integral. Trees minimizing the energy under various additional conditions have been determined in the past, e.g., trees with a given diameter or trees with a perfect matching. However, it is quite a natural problem to minimize the energy of trees with bounded maximum degree--clearly, the case of maximum degree 4 (so-called chemical trees) is the most important one. We will show that the trees with given maximum degree that minimize the energy are the same that have been shown previously to minimize the Hosoya index and maximize the Merrifield-Simmons index, thus also proving a conjecture due to Fischermann et al. Finally, we show that the minimal energy grows linearly with the size of the trees, with explicitly computable growth constants that only depend on the maximum degree.
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