具有相反奇偶性指数的整循环图的能量最大化
Energy Maximisation for Integral Circulant Graphs with Opposite-Parity Exponents
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中文总结 AI 辅助
该研究确定了阶为p^(2r)·q^(2s)+1的整循环图的最大能量，证明了Roldan棋盘除数集的猜想最大化性，得到了显式闭式能量公式，核心方法为加权素幂Ramanujan变换的半定奇偶性定理。
中文摘要 AI 辅助
对于有限图，其能量为邻接特征值绝对值的和。设p和q为不同的奇素数且q≥5，r≥1，s≥0。我们确定了所有阶为p^(2r)·q^(2s)+1的整循环图中的最大能量。唯一的能量最大化除数集是棋盘集，由所有满足i+j为偶数的除数p^i q^j组成，其中i∈[0,2r]，j∈[0,2s+1]，我们得到了对应最大能量的显式闭式公式。特别地，当q≥5且r=s=1时，我们的定理证明了Roldan棋盘除数集的猜想最大化性，并得到了他的闭式能量公式。核心方法是通过与奇偶性无关的同余约化和块Schur递推证明的加权素幂Ramanujan变换的半定奇偶性定理，对除数矩阵中心化后，从Kronecker积算子的半定界得到严格符号矩阵不等式，对等性分析确定了棋盘模式并证明了唯一性。
英文摘要
For a finite graph, its energy is the sum of the absolute values of its adjacency eigenvalues. Let p and q be distinct odd primes with q at least 5, and let r be at least 1 and s be nonnegative. We determine the maximum energy among all integral circulant graphs whose order is p to the power 2r times q to the power 2s plus 1. The unique energy-maximising divisor set is the checkerboard set consisting of all divisors p raised to the power i times q raised to the power j, where i ranges from zero to 2r, j ranges from zero to 2s plus 1, and i plus j is even, and we obtain an explicit closed formula for the corresponding maximum energy. In particular, for q at least 5, when r and s are both equal to 1, our theorem establishes the conjectured maximality of Roldan's checkerboard divisor set and recovers his closed-form energy formula. The main ingredient is a semidefinite parity theorem for weighted prime-power Ramanujan transforms, proved by a parity-independent congruence reduction and a block Schur recurrence. Centring the divisor matrix then yields a sharp sign-matrix inequality from semidefinite bounds for a Kronecker-product operator. Analysing equality identifies the checkerboard pattern and proves uniqueness.
发表机构
- School of Mathematics and Statistics, Weifang University（潍坊学院数学与统计学院）
- Weifang, Shandong, China（中国山东潍坊）
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