具有规定最大度的极大σ-不规则树的完整刻画
A complete characterization of maximally σ-irregular trees with prescribed maximum degree
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中文总结 AI 辅助
本文刻画了n≥Δ(Δ-1)+1时所有Δ≥6的极大σ-不规则树，引入三类树推导其σ-不规则性公式，证明极大树对应达到最大σ-不规则性的类，且Δ≥7时T'''类树的σ-不规则性更大。
中文摘要 AI 辅助
图G=(V,E)的σ-不规则性定义为对所有边uv∈E，求和(d(u)-d(v))²，其中d(u)表示顶点u的度。顶点数为n、最大度为Δ的树，若在所有此类树中达到最大σ-不规则性，则称为极大树。目前已确定化学树（Δ≤4）和Δ=5时的极大树。本文中，当n≥Δ(Δ-1)+1时，我们对所有Δ≥6的极大树进行了刻画。我们引入三类树：T'{n,Δ}、T''{n,Δ}和T'''{n,Δ}。同一类内所有树的σ-不规则性相同，且我们推导了每类树达到的σ-不规则性的显式公式。通过比较这些公式，可确定给定n和Δ时哪类树的σ-不规则性最大。随后我们证明，一棵树是极大树当且仅当它属于达到该最大σ-不规则性的类。与Δ≤5的情况不同，自然候选类T'{n,Δ}和T''{n,Δ}并不充分，因为对于每个Δ≥7和每个n≡3 mod Δ，T'''{n,Δ}中的树具有严格更大的σ-不规则性。
英文摘要
The sigma-irregularity of a graph G = (V, E) is defined as the sum, over all edges uv in E, of (d(u) - d(v))^2, where d(u) denotes the degree of vertex u. A tree on n vertices with maximum degree Delta is called maximal if it attains the greatest possible sigma-irregularity among all such trees. The maximal trees are already known for chemical trees (Delta <= 4) and for Delta = 5. In this paper, we characterize the maximal trees for every Delta >= 6 when n >= Delta(Delta - 1) + 1. We introduce three families of trees, T'{n,Delta}, T''{n,Delta}, and T'''{n,Delta}. All trees within the same family have the same sigma-irregularity, and we derive an explicit formula for the value attained by each family. Comparing these formulas determines which family has the greatest sigma-irregularity for given n and Delta. We then prove that a tree is maximal if and only if it belongs to a family attaining this greatest value. In contrast to the case Delta <= 5, the two natural candidate families T'{n,Delta} and T''{n,Delta} are not sufficient, since for every Delta >= 7 and every n congruent to 3 modulo Delta, the trees in T'''{n,Delta} have strictly greater sigma-irregularity.
