首页/论文/2006.08892

arXiv 2006.088922020-06-16math.CO

The maximal tree with respect to the exponential of the second Zagreb index

Mingyao Zeng, Hanyuan Deng

英文摘要：

The second Zagreb index is $M_2(G)=\sum_{uv\in E(G)}d_{G}(u)d_{G}(v)$. It was found to occur in certain approximate expressions of the total $π$-electron energy of alternant hydrocarbons and used by various researchers in their QSPR and QSAR studies. Recently the exponential of a vertex-degree-based topological index was introduced. It is known that among all trees with $n$ vertices, the exponential of the second Zagreb index $e^{M_2}$ attains its minimum value in the path $P_n$. In this paper, we show that $e^{M_2}$ attains its maximum value in the balanced double star with $n$ vertices and solve an open problem proposed by Cruz and Rada [R. Cruz, J. Rada, The path and the star as extremal values of vertex-degree-based topological indices among trees, MATCH Commun. Math. Comput. Chem. 82 (3) (2019) 715-732].

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Sharp upper bounds for multiplicative Zagreb indices of bipartite graphs with given diameter

The first multiplicative Zagreb index of a graph $G$ is the product of the square of every vertex degree, while the second multiplicative Zagreb index is the product of the degree of each edge over all edges. In our work, we explore the multiplicative Zagreb indices of bipartite graphs of order $n$ with diameter $d$, and sharp upper bounds are obtained for these indices of graphs in $\mathcal{B}(n,d)$, where $\mathcal{B}(n, d)$ is the set of all $n$-vertex bipartite graphs with the diameter $d$. In addition, we explore the relationship between the maximal multiplicative Zagreb indices of graph…

2017-04-27math.CO

Maximum first Zagreb index of orientations of unicyclic graphs with given matching number

Let $D=(V,A)$ be a digraphs without isolated vertices. The first Zagreb index of a digraph $D$ defined as a summation over all arcs, $M_1(D)=\frac{1}{2}\sum\limits_{uv\in A}(d^{+}_{u}+d^{-}_v)$, where $d^{+}_u$(resp. $d^{-}_u$) denotes the out-degree (resp. in-degree) of the vertex $u$. In this paper, we give the maximal values and maximal digraphs of first Zagreb index over the set of all orientations of unicyclic graphs with $n$ vertices and matching number $m$ $(2\leq m\leq \lfloor \frac{n}{2}\rfloor)$.

2022-05-29math.COmath.OC

具有最大补第二Zagreb指数的极图

Extremal graphs with maximum complementary second Zagreb index

本文证实了Furtula和Oz关于阶为$n$且具有最大补第二Zagreb指数的极大图的猜想。

2025-03-03math.CO

树和二部图的指数超 Zagreb 指数与结构性质

The Exponential Hyper-Zagreb Indices and Structural Properties of Trees and Bipartite Graphs

本文研究树和二部图的结构性质，聚焦超Zagreb指数的极值性，推广至指数形式，结合度序列优化等方法分析树的结构，探讨二部图的公平着色等问题，提出马尔可夫链算法，为图论及相关应用提供理论支持。

2025-08-19math.CO

Majorization under constraints and bounds of the second Zagreb index

In this paper we present a theoretical analysis in order to establish maximal and minimal vectors with respect to the majorization order of particular subsets of \Re ^n: Afterwards we apply these issues to the calcula- tion of bounds for a topological descriptor of a graph known as the second Zagreb index. Finally, we show how our bounds may improve the re- sults obtained in the literature, providing some theoretical and numerical examples.

2011-05-10math.COmath.OC

关于树的减少第二Zagreb指数下界进一步结果

Further results on the lower bound on reduced Zagreb index of trees

University of Niš（尼什大学）；Meta Inc（Meta公司）

本文改进了树中减少第二Zagreb指数的下界结果，针对λ≥-1的情况，通过两种方法确定了Δ=3和Δ=4时λ=-2的最小值，并刻画了极值树。

2026-04-07cs.DMmath.CO

更多相关论文收起更多论文

聚多米诺骨牌链上的最大增广Zagreb指数

Maximum augmented Zagreb index on polyomino chains

本文提出动态规划方法识别极值聚多米诺骨牌链，线性时间内计算并枚举极大配置，解决了2016年关于最大化增广Zagreb指数的问题，并推导了闭式表达式。

2025-09-12math.CO

New bounds of degree-based topological indices for some classes of $c$-cyclic graphs

Making use of a majorization technique for a suitable class of graphs, we derive upper and lower bounds for some topological indices depending on the degree sequence over all vertices, namely the first general Zagreb index and the first multiplicative Zagreb index. Specifically, after characterizing $c-$cyclic graphs $(0\leq c\leq 6)$ as those whose degree sequence belongs to particular subsets of $\mathbb{R}^{n}$, we identify the maximal and minimal vectors of these subsets with respect to the majorization order. This technique allows us to determine lower and upper bounds of the above indice…

2013-11-22math.COmath.OC

On the conjecture about the exponential reduced Sombor index

Let $G=(V(G),E(G))$ be a graph and $d(v)$ be the degree of the vertex $v\in V(G)$. The exponential reduced Sombor index of $G$, denoted by $e^{SO_{red}}(G)$, is defined as $$e^{SO_{red}}(G)=\sum_{uv\in E(G)}e^{\sqrt{(d(u)-1)^2+(d(v)-1)^2}}.$$ We obtain a characterization of extremal trees with the maximal exponential reduced Sombor index among all chemical trees of order $n$. This result shows the conjecture on the exponential reduced Sombor index proposed by Liu, You, Tang and Liu [On the reduced Sombor index and its applications, MATCH Commun. Math. Comput. Chem. 86 (2021) 729--753] is negat…

2022-09-02math.CO

Dimensionally Exponential Lower Bounds on the $L^p$ Norms of the Spherical Maximal Operator for Cartesian Powers of Finite Trees and Related Graphs

Let $T$ be a finite tree graph, $T^N$ be the Cartesian power graph of $T$, and $d^N$ be the graph distance metric on $T^N$. Also let \[ \mathbb S_r^N(x) := \{v \in T^N: d^N(x,v) = r\} \] be the sphere of radius $r$ centered at $x$ and $M$ be the spherical maximal averaging operator on $T^N$ given by \[ Mf(x) := \sup_{\substack{r \geq 0 \\ \mathbb S_r^N(x) \neq \emptyset}} \frac{1}{|\mathbb S_r^N(x)|} |\sum_{\mathbb S_r^N(x)} f(y)|. \] We will show that for any fixed $1 \leq p \leq \infty$, the $L^p$ operator norm of $M$, i.e. \[ \|M\|_p := \sup_{\|f\|_p = 1} \|Mf\|_p, \] grows exponentially in…

2015-09-09cs.DMmath.CAmath.CO