Ya-Hong Chen, Xiao-Dong Zhang
In this paper, we investigate how the Wiener index of unicyclic graphs varies with graph operations. These results are used to present a sharp lower bound for the Wiener index of unicyclic graphs of order $n$ with girth and the matching number $β\ge \frac{3g}{2}$. Moreover, we characterize all extremal graphs which attain the lower bound.
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On minimizing the Wiener index of unicyclic graphs with fixed girth and given degree sequence
针对固定围长与给定度序列的单圈图，研究依据质心位置提出三类 Wiener 指数最小化候选结构。
In this paper we consider the following problem: Over the class of all simple connected unicyclic graphs on $n$ vertices with girth $g$ ($n,g$ being fixed), which graph minimizes the Laplacian spectral radius? We prove that the graph $U_{n,g}$ (defined in Section 1) uniquely minimizes the Laplacian spectral radius for $n\geq 2g-1$ when $g$ is even and for $n\geq 3g-1$ when $g$ is odd.
Unicyclic Graphs of Arbitrary Girth with the Same Chromatic Symmetric Function
研究图的色对称函数能否区分非同构单圈图，通过应用“三重删除”模关系，展示了一系列围长递增且具有相同色对称函数的非同构连通单圈图对，还给出首个具有相同色对称函数的无限二分图族。
In this article, we determine the maximum Wiener indices of unicyclic graphs with given number of vertices and matching number. We also characterize the extremal graphs. This solves an open problem of Du and Zhou.
The aim of this paper is to obtain new inequalities for a large family of generalizations of the Wiener Index and to characterize the set of extremal graphs with respect to them. Our main results provide upper and lower bounds for these topological indices on unicyclic graphs.
The \emph{Wiener index} is a widely studied topological index of graphs. One of the main problems in the area is to determine which graphs of given properties attain the extremal values of Wiener index. In this paper we resolve an open problem posed by Du in 2012. To this end we determine the unicyclic bipartite graphs with given size of parts having the maximum Wiener index. This completes the previous research in which the minimum case was solved.
Let $G$ be a graph of order $n$ and let $\mathcal{L}(G,λ)=\sum_{k=0}^n (-1)^{k}c_{k}(G)λ^{n-k}$ be the characteristic polynomial of its Laplacian matrix. Motivated by Ilić and Ilić's conjecture [A. Ilić, M. Ilić, Laplacian coefficients of trees with given number of leaves or vertices of degree two, Linear Algebra and its Applications 431(2009)2195-2202.] on all extremal graphs which minimize all the Laplacian coefficients in the set $\mathcal{U}_{n,l}$ of all $n$-vertex unicyclic graphs with the number of leaves $l$, we investigate properties of the minimal elements in the partial set $(\mathc…
Let $G=(V(G),E(G))$ be a molecular graph, where $V(G)$ and $E(G)$ are the sets of vertices (atoms) and edges (bonds). A topological index of a molecular graph is a numerical quantity which helps to predict the chemical/physical properties of the molecules. The Wiener, Wiener polarity and the terminal Wiener indices are the distance based topological indices. In this paper, we described a linear time algorithm {\bf(LTA)} that computes the Wiener index for acyclic graphs and extended this algorithm for unicyclic graphs. The same algorithms are modified to compute the terminal Wiener index and th…
On the maximum $A_α$-spectral radius of unicyclic and bicyclic graphs with fixed girth or fixed number of pendant vertices
本文研究了单圈图和双圈图的 $A_α$-谱半径极值问题，分别刻画了在固定围长和固定悬挂点数条件下具有最大 $A_α$-谱半径的图。
The reverse degree distance is a connected graph invariant closely related to the degree distance proposed in mathematical chemistry. We determine the unicyclic graphs of given girth, number of pendant vertices and maximum degree, respectively, with maximum reverse degree distances.
The strong spectral property for some families of unicyclic graphs
本文研究具有强谱性质的图类，完全刻画了围长为三的单圈图，并证明围长至多五的蝌蚪图具有该性质而围长六的蝌蚪图不具有。
Let $G$ be an unicyclic graph of order $n$ and let $Q_G(x)= det(xI-Q(G))={matrix} \sum_{i=1}^n (-1)^i φ_i x^{n-i}{matrix}$ be the characteristic polynomial of the signless Laplacian matrix of a graph $G$. We give some transformations of $G$ which decrease all signless Laplacian coefficients in the set $\mathcal{G}(n,m)$. $\mathcal{G}(n,m)$ denotes all n-vertex unicyclic graphs with matching number $m$. We characterize the graphs which minimize all the signless Laplacian coefficients in the set $\mathcal{G}(n,m)$ with odd (resp. even) girth. Moreover, we find the extremal graphs which have mini…
Laplacian eigenvalue distribution for unicyclic graphs
该研究针对单圈图，以图的直径和围长为参数，给出了区间[0,1)内其拉普拉斯特征值数量的上界，推进了单圈图谱性质相关研究。
In the study of topological indices two negative correlations are well known: that between the number of subtrees and the Wiener index (sum of distances), and that between the Merrifield-Simmons index (number of independent vertex subsets) and the Hosoya index (number of independent edge subsets). That is, among a certain class of graphs, the extremal graphs that maximize one index usually minimize the other, and vice versa. In this paper, we first study the numbers of subtrees in unicyclic graphs and unicyclic graphs with a given girth, further confirming its opposite behavior to the Wiener i…
Cut vertices are often used as a measure of nodes' importance within a network. They are those nodes whose failure disconnects a graph. Let N(G) be the number of connected induced subgraphs of a graph $G$. In this work, we investigate the maximum of N(G) where $G$ is a unicyclic graph with $n$ nodes of which $c$ are cut vertices. For all valid $n,c$, we give a full description of those maximal (that maximise N(.)) unicyclic graphs. It is found that there are generally two maximal unicyclic graphs. For infinitely many values of $n,c$, however, there is a unique maximal unicyclic graph with $n$ …
An oriented graph $G^σ$ is a digraph without loops and multiple arcs, where $G$ is called the underlying graph of $G^σ$. Let $S(G^σ)$ denote the skew-adjacency matrix of $G^σ$. The rank of the skew-adjacency matrix of $G^σ$ is called the {\it skew-rank} of $G^σ$, denoted by $sr(G^σ)$. The skew-adjacency matrix of an oriented graph is skew symmetric and the skew-rank is even. In this paper we consider the skew-rank of simple oriented graphs. Firstly we give some preliminary results about the skew-rank. Secondly we characterize the oriented graphs with skew-rank 2 and characterize the oriented g…
Let $G_w$ be a weighted graph. The \textit{inertia} of $G_w$ is the triple $In(G_w)=\big(i_+(G_w),i_-(G_w), $ $ i_0(G_w)\big)$, where $i_+(G_w),i_-(G_w),i_0(G_w)$ are the number of the positive, negative and zero eigenvalues of the adjacency matrix $A(G_w)$ of $G_w$ including their multiplicities, respectively. $i_+(G_w)$, $i_-(G_w)$ is called the \textit{positive, negative index of inertia} of $G_w$, respectively. In this paper we present a lower bound for the positive, negative index of weighted unicyclic graphs of order $n$ with fixed girth and characterize all weighted unicyclic graphs att…
In this paper we investigate the hypergraphs whose spectral radii attain the maximum among all uniform hypergraphs with given number of edges. In particular we characterize the hypergraph(s) with maximum spectral radius over all unicyclic hypergraphs, linear or power unicyclic hypergraphs with given girth, linear or power bicyclic hypergraphs, respectively.
The square of a connected graph $G$ is obtained from $G$ by adding an edge between every pair of vertices at distance $2$. In this paper we give some upper or lower bounds for the spectral radius of the square of connected graphs, trees and unicyclic graphs respectively.We also investigate the spectral radius of the square of unicyclic graphs with given girth or trees with fixed diameter.
Independent Locating-Dominating Sets in Pseudotrees
研究伪树中独立定位支配集的存在性，证明girth为4的图可能无ILD集，且在树和单环图中提供ILD集的存在条件及算法。
Over all graphs (or unicyclic graphs) of a given order, we characterise those graphs that minimise or maximise the number of connected induced subgraphs. For each of these classes, we find that the graphs that minimise the number of connected induced subgraphs coincide with those that are known to maximise the Wiener index (the sum of the distances between all unordered pairs of vertices), and vice versa. For every $k$, we also determine the connected graphs that are extremal with respect to the number of $k$-vertex connected induced subgraphs. We show that, in contrast to the minimum which is…
The energy of a graph $G$, denoted by $E(G)$, is defined as the sum of the absolute values of all eigenvalues of $G$. Let $n$ be an even number and $\mathbb{U}_{n}$ be the set of all conjugated unicyclic graphs of order $n$ with maximum degree at most $3$. Let $S_n^{\frac{n}{2}}$ be the radialene graph obtained by attaching a pendant edge to each vertex of the cycle $C_{\frac{n}{2}}$. In [Y. Cao et al., On the minimal energy of unicyclic Hückel molecular graphs possessing Kekulé structures, Discrete Appl. Math. 157 (5) (2009), 913--919], Cao et al. showed that if $n\geq 8$, $S_n^{\frac{n}{2}}\…
On the third ABC index of trees and unicyclic graphs
本文研究树与单圈图的第三原子键连通性指数，分别确定给定直径的树和给定围长的单圈图的最大 $ABC_3$ 指数，并刻画达到极值的图结构。
The trace of uniform hypergraphs with application to Estrada index
本文研究超图邻接张量的迹，给出超树和线性单圈超图的迹的新表达式及割点扰动结果，并据此确定固定边数和完美匹配下Estrada指标最大的唯一超树，以及固定边数和围长3下Estrada指标最大的唯一单圈超图。
The Wiener index of a connected graph is the sum of the distance of all pairs of distinct vertices. It was introduced by Wiener in 1947 to analyze some aspects of branching by fitting experimental data for several properties of alkane compounds. Denote by $\mathcal{U}_{n,r}$ the set of unicyclic graphs with $n$ vertices and $r$ vertices of even degree. In this paper we present a structural result on the graphs in $\mathcal{U}_{n,r}$ with minimum Wiener index and completely characterize such graphs when $ r\leq \frac{n+3}{2}$.
For a graph $G,$ we denote the number of connected subgraphs of $G$ by $F(G)$. For a tree $T$, $F(T)$ has been studied extensively and it has been observed that $F(T)$ has a reverse correlation with Wiener index of $T$. Based on that, we call $F(G),$ the core index of $G$. In this paper, we characterize the graphs which extremize the core index among all graphs on $n$ vertices with $k\geq 0$ connected components. We extend our study of core index to unicyclic graphs and connected graphs with fixed number of pendant vertices. We obtained the unicyclic graphs which extremize the core index over …
A graph G is called well-indumatched if all of its maximal induced matchings have the same size. In this paper we characterize all well-indumatched trees. We provide a linear time algorithm to decide if a tree is well-indumatched or not. Then, we characterize minimal well-indumatched graphs of girth at least 9 and show subsequently that for an odd integer g greater than or equal to 9 and different from 11, there is no well-indumatched graph of girth g. On the other hand, there are infinitely many well-indumatched unicyclic graphs of girth k, where k is in {3, 5, 7} or k is an even integer grea…
Discrete Einstein metrics on unicyclic graphs
研究单圈图上离散爱因斯坦度量，将有限树相关理论扩展至此。确定树状情形持续条件及范围，超出范围问题呈分段线性，有新现象。证明正则太阳图的爱因斯坦度量存在且唯一。
We prove a conjecture of Nadjafi-Arani, Khodashenas and Ashrafi on the difference between the Szeged and Wiener index of a graph. Namely, if $G$ is a 2-connected non-complete graph on $n$ vertices, then $Sz(G)-W(G)\ge 2n-6$. Furthermore, the equality is obtained if and only if $G$ is the complete graph $K_{n-1}$ with an extra vertex attached to either $2$ or $n-2$ vertices of $K_{n-1}$. We apply our method to strengthen some known results on the difference between the Szeged and Wiener index of bipartite graphs, graphs of girth at least five, and the difference between the revised Szeged and W…
The quotients between the (revised) Szeged index and Wiener index of graphs
本文刻画了单圈图和双圈图中Wiener指数极值图，并基于此研究了（修正）Szeged指数与Wiener指数之商的下界及极值图。
The Graovac-Pisanski index, also called the modified Wiener index, was introduced in 1991 and represents an extension of the original Wiener index, because it considers beside the distances in a graph also its symmetries. Similarly as Wiener in 1947 showed the correlation of the Wiener indices of the alkane series with the boiling points, in 2018 the connection between the Graovac-Pisanski index and the melting points of some hydrocarbon molecules was established. In this paper, we prove that the Graovac-Pisanski index of any connected bipartite graph as well as of any connected graph on an ev…
On the sum of eccentric subgraph numbers of the vertices of a connected graph
针对连通图顶点的偏心子图数之和，定义了顶点的偏心子图数与图的偏心子图指数，研究了单圈图、树等特定图类中使偏心子图指数取最大值的图结构。
Resolving Open Problems on the Euler Sombor Index
本文针对 Khanra 和 Das 提出的 Euler Sombor 指数开放问题，确定固定阶数与围长的单圈图及连通图的最小 $EUS$ 与极值结构，并分类固定阶数和叶子数时的最大 $EUS$ 连通图。
Hansen et. al. used the computer programm AutoGraphiX to study the differences between the Szeged index $Sz(G)$ and the Wiener index $W(G)$, and between the revised Szeged index $Sz^*(G)$ and the Wiener index for a connected graph $G$. They conjectured that for a connected nonbipartite graph $G$ with $n \geq 5$ vertices and girth $g \geq 5,$ $ Sz(G)-W(G) \geq 2n-5. $ Moreover, the bound is best possible as shown by the graph composed of a cycle on 5 vertices, $C_5$, and a tree $T$ on $n-4$ vertices sharing a single vertex. They also conjectured that for a connected nonbipartite graph $G$ with …
Denote by $L_{g, l}$ the $lollipop$ $graph$ obtained by attaching a pendant path $\mathbb{P}=v_{g}v_{g+1}\cdots v_{g+l}$ ($l\geq 1$) to a cycle $\mathbb{C}=v_{1}v_{2}\cdots v_{g}v_{1}$ ($g\geq 3$). A $\mathcal {F}_{g, l}$-$graph$ of order $n\geq g+1$ is defined to be the graph obtained by attaching $n-g-l$ pendent vertices to some of the nonpendant vertices of $L_{g, l}$ in which each vertex other than $v_{g+l-1}$ is attached at most one pendant vertex. A $\mathcal {F}^{\circ}_{g, l}$-graph is a $\mathcal {F}_{g, l}$-$graph$ in which $v_{g}$ is attached with pendant vertex. Denote by $q_{min}$…
A graph is hypohamiltonian if it is not Hamiltonian, but the deletion of any single vertex gives a Hamiltonian graph. Until now, the smallest known planar hypohamiltonian graph had 42 vertices, a result due to Araya and Wiener. That result is here improved upon by 25 planar hypohamiltonian graphs of order 40, which are found through computer-aided generation of certain families of planar graphs with girth 4 and a fixed number of 4-faces. It is further shown that planar hypohamiltonian graphs exist for all orders greater than or equal to 42. If Hamiltonian cycles are replaced by Hamiltonian pat…
The subpath number of cactus graphs
本文研究仙人掌图的子路径数，刻画了具有n个顶点和k个圈的极大与极小仙人掌图结构，并证明子路径数与子树数及Wiener指数均不相关。
Maker-Breaker Sabotage Game
研究在图上进行的Maker-Breaker破坏游戏，通过策略分析给出树的破坏数多项式时间计算公式，并得到单圈图、无桥次立方图等图类的破坏数界或精确值。
Structural Properties and Applications of the Augmented Sombor Index
本文定义并研究了增强Sombor指数，建立了其极值界并刻画了极图，包括单圈图、树、二部图和化学图，并通过QSPR分析验证了其化学适用性。
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