Karthik S, M Sabeel K, K. Paramasivam
In this article, some necessary conditions of group vertex magicness of graphs with at least one pendant, group vertex magicness of product graphs, are proved. A characterization of group vertex magicness of trees of diameter up to 5. for all infinite Abelian groups with finitely many torsion elements, is also obtained.
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Group Vertex Magicness of H-join and Generalised Friendship Graph
本文研究了图族的H-join和广义友谊图的群顶点魔性。
In this paper, we give a characterization of unicyclic graphs with diameter at most 4 which are A-vertex magic. Moreover, let G be a bicyclic graph of diameter 3, then G is group vertex magic if and only if G = M11(0, 0).
In this paper, we provide few results on the group distance magic labeling of lexicographic product and direct product of two graphs. We also prove some necessary conditions for a graph to be group distance magic and provide a characterization for a tree to be group distance magic.
Prüfer codes on vertex-colored rooted trees
本文扩展Prüfer编码至顶点着色有根树，定义规范标记与VCPC，证明其可判定树同构、子树关系及子式同构，应用于货运网络分解。
Unique paired vs edge-vertex minimum dominating sets in trees
本文证明了树中唯一最小边顶支配集与唯一最小配对支配集的等价性。
In 1997 Klavžar and Gutman suggested a generalization of the Wiener index to vertex-weighted graphs. We minimize the Wiener index over the set of trees with the given vertex weights' and degrees' sequences and show an optimal tree to be the, so-called, Huffman tree built in a bottom-up manner by sequentially connecting vertices of the least weights.
Koh and Tay introduced a new family of graphs, $G$ vertex-multiplications, as an extension of complete $n$-partite graphs. They proved a fundamental classification of $G$ vertex-multiplications into three classes $\mathscr{C}_0, \mathscr{C}_1$ and $\mathscr{C}_2$. It was shown that any vertex-multiplication of a tree with diameter at least 3 does not belong to the class $\mathscr{C}_2$. Furthermore, for vertex-multiplications of trees with diameter $5$, some necessary and sufficient conditions for $\mathscr{C}_0$ were established. In this paper, we give a complete characterisation of vertex-mu…
Given a tree of weighted vertices, it is sometimes possible to break the tree into two equally-weighted subtrees within an allowable error. We give a fast algorithm that finds an edge which breaks the tree into equal-weight components or determines there is no such edge.
A \emph{$k$-tree} is a chordal graph with no $(k+2)$-clique. An \emph{$\ell$-tree-partition} of a graph $G$ is a vertex partition of $G$ into `bags', such that contracting each bag to a single vertex gives an $\ell$-tree (after deleting loops and replacing parallel edges by a single edge). We prove that for all $k\geq\ell\geq0$, every $k$-tree has an $\ell$-tree-partition in which every bag induces a connected $\floor{k/(\ell+1)}$-tree. An analogous result is proved for oriented $k$-trees.
We prove that, if $m$ is sufficiently large, every graph on $m+1$ vertices that has a universal vertex and minimum degree at least $\lfloor \frac{2m}{3} \rfloor$ contains each tree $T$ with $m$ edges as a subgraph. Our result confirms, for large $m$, an important special case of a conjecture by Havet, Reed, Stein, and Wood. The present paper builds on the results of a companion paper in which we proved the statement for all trees having a vertex that is adjacent to many leaves.
The Steiner $k$-eccentricity of a vertex in graph $G$ is the maximum Steiner distance over all $k$-subsets containing the vertex. %Some general properties of the Steiner 3-eccentricity of trees are given. Let $\mathbb{T}_n$ be the set of all $n$-vertex trees, $\mathbb{T}_{n,Δ}$ be the set of $n$-vertex trees with given maximum degree equal to $Δ$, $\mathbb{T}_n^k$ be the set of $n$-vertex trees with exactly $k$ vertices of maximum degree, and let $\mathbb{T}_{n,Δ}^k$ be the set of $n$-vertex trees with exactly $k$ vertices of given maximum degree equal to $Δ.$ In this paper, we first determine…
A vertex cover of a graph $G = (V, E)$ is a set $X \subseteq V$ such that each edge of $G$ is incident to at least one vertex of $X$. A dominating set $D \subseteq V$ is a total dominating set of $G$ if the subgraph induced by $D$ has no isolated vertices. A $(γ_t-τ)$-set of $G$ is a minimum vertex cover which is also a minimum total dominating set. In this article we give a constructive characterization of trees having a $(γ_t-τ)$-set.
Koh and Tay proved a fundamental classification of $G$ vertex-multiplications into three classes $\mathscr{C}_0, \mathscr{C}_1$ and $\mathscr{C}_2$. They also showed that any vertex-multiplication of a tree with diameter at least 3 does not belong to the class $\mathscr{C}_2$. Of interest, $G$ vertex-multiplications are extensions of complete $n$-partite graphs and Gutin characterised complete bipartite graphs with an ingenious use of Sperner's Theorem. In this paper, we investigate vertex-multiplications of trees with diameter $4$ in $\mathscr{C}_0$ (or $\mathscr{C}_1$) and exhibit its intric…
Let $G=(V_G, E_G)$ be a simple connected graph. The eccentric distance sum of $G$ is defined as $ξ^{d}(G) = \sum_{v\in V_G}\varepsilon_{G}(v)D_{G}(v)$, where $\varepsilon_G(v)$ is the eccentricity of the vertex $v$ and $D_G(v) = \sum_{u\in V_G}d_G(u,v)$ is the sum of all distances from the vertex $v$. In this paper the tree among $n$-vertex trees with domination number $γ$ having the minimal eccentric distance sum is determined and the tree among $n$-vertex trees with domination number $γ$ satisfying $n = kγ$ having the maximal eccentric distance sum is identified, respectively, for $k=2,3,\fr…
In this paper and a companion paper, we prove that, if $m$ is sufficiently large, every graph on $m+1$ vertices that has a universal vertex and minimum degree at least $\lfloor \frac{2m}{3} \rfloor$ contains each tree $T$ with $m$ edges as a subgraph. Our result confirms, for large $m$, an important special case of a recent conjecture by Havet, Reed, Stein, and Wood. The present paper already contains an approximate version of the result.
Over some types of trees with a given number of vertices, which trees minimize or maximize the total number of subtrees or leaf containing subtrees are studied. Here are some of the main results:\ (1)\, Sharp upper bound on the total number of subtrees (resp. leaf containing subtrees) among $n$-vertex trees with a given matching number is determined; as a consequence, the $n$-vertex tree with domination number $γ$ maximizing the total number of subtrees (resp. leaf containing subtrees) is characterized. (2)\, Sharp lower bound on the total number of leaf containing subtrees among $n$-vertex tr…
There is a unique path from the root of a tree to any other vertex. Every vertex, except the root, has a parent: the adjoining vertex on this unique path. This is the conventional definition of the parent vertex. For complete binary trees, however, we show that it is useful to define another parent vertex, called a \emph{distant parent}. The study of distant parents leads to novel connections with dyadic rational numbers. Moreover, we apply the concepts of close and distant parent vertices to deduce an apparently new sense in which continued fractions are `best' rational approximations.
The vertex v of a graph G is called a 1-critical-vertex for the maximum genus of the graph, or for simplicity called 1-critical-vertex, if G-v is a connected graph and °M(G - v) = °M(G) - 1. In this paper, through the joint-tree model, we obtained some types of 1-critical-vertex, and get the upper embeddability of the Spiral Snm .
Limbs and Cospectral Vertices in Trees
该研究扩展了Schwenk的结果，证明了在加权树中，几乎所有的树都包含给定的肢体，并且随着顶点数增加，共谱顶点的比例趋近于1。
In this paper we at first consider plane trees with the root vertex and a marked directed edge, outgoing from the root vertex. For such trees we introduce a new characteristic --- the \emph{parity}, using the bracket code. It turns out that the parity depends only on the root vertex (not on the marked edge). And in the case of an even number of vertices the parity does not depend on the root vertex also. Then we consider rotation groups of bipartite trees, study their properties and prove that in the case of even number of vertices rotation groups of even and odd trees are different.
A set D of vertices of a graph G with vertex set V is irredundant if each non-isolated vertex of G[D] has a neighbour in V-D that is not adjacent to any other vertex in D. The upper irredundance number IR(G) is the largest cardinality of an irredundant set of G; an IR(G)-set is an irredundant set of cardinality IR(G). The IR-graph of G has the IR(G)-sets as vertex set, and sets A and B are adjacent if and only if B can be obtained from A by exchanging a single vertex of A for an adjacent vertex in B. An IR-tree is an IR-graph that is a tree. We characterize IR-trees of diameter 3 by showing th…
A tree $T$ in an edge-colored graph $H$ is called a \emph{monochromatic tree} if all the edges of $T$ have the same color. For $S\subseteq V(H)$, a \emph{monochromatic $S$-tree} in $H$ is a monochromatic tree of $H$ containing the vertices of $S$. For a connected graph $G$ and a given integer $k$ with $2\leq k\leq |V(G)|$, the \emph{$k$-monochromatic index $mx_k(G)$} of $G$ is the maximum number of colors needed such that for each subset $S\subseteq V(G)$ of $k$ vertices, there exists a monochromatic $S$-tree. In this paper, we prove that for any connected graph $G$, $mx_k(G)=|E(G)|-|V(G)|+2$ …
Let v(G) be the number of vertices and t(G,k) the maximum number of disjoint k-edge trees in G. In this paper we show that (a1) if G is a graph with every vertex of degree at least two and at most s, where s > 3, then t(G,2) is at least v(G)/(s+1), (a2) if G is a graph with every vertex of degree at least two and at most 3 and G has no 5-vertex components, then t(G,2) is at least v(G)/4, (a3) if G is a graph with every vertex of degree at least one and at most s and G has no k--vertex component, where k >1 and s > 2, then t(G,k) is at least (v(G) - k)/(sk - k +1), and (a4) the above bounds are…
We prove that there exists $C>0$ such that any $(n+Ck)$-vertex tournament contains a copy of every $n$-vertex oriented tree with $k$ leaves, improving the previously best known bound of $n+O(k^2)$ vertices to give a result tight up to the value of $C$. Furthermore, we show that, for each $k$, there exists $n_0$, such that, whenever $n\geqslant n_0$, any $(n+k-2)$-vertex tournament contains a copy of every $n$-vertex oriented tree with at most $k$ leaves, confirming a conjecture of Dross and Havet.
We prove that the chromatic symmetric function of any $n$-vertex tree containing a vertex of degree $d\geq \log _2n +1$ is not $e$-positive, that is, not a positive linear combination of elementary symmetric functions. Generalizing this, we also prove that the chromatic symmetric function of any $n$-vertex connected graph containing a cut vertex whose deletion disconnects the graph into $d\geq\log _2n +1$ connected components is not $e$-positive. Furthermore we prove that any $n$-vertex bipartite graph, including all trees, containing a vertex of degree greater than $\lceil \frac{n}{2}\rceil$ …
Tree-independence number and forbidden induced subgraphs: excluding a $6$-vertex path and a $(2,t)$-biclique
研究排除6顶点路径和(2,t)-双 clique的图的树独立数上限，推进Dallard等人猜想的进展。
Let $\mathcal{CT}_{n,k}$ and $\mathcal{CT}^*_{n,b}$ be the classes of all $n$-vertex chemical trees with $k$ segments and $b$ branching vertices, respectively, where $3\le k\le n-1$ and $1\le b< \frac{n}{2}-1$. The solution of the problem of finding trees from the class $\mathcal{CT}_{n,k}$ or $\mathcal{CT}^*_{n,b}$, with the minimum first Zagreb index or minimum second Zagreb index follows directly from the main results of [MATCH Commun. Math. Comput. Chem. 72 (2014) 825-834] or [MATCH Commun. Math. Comput. Chem. 74 (2015) 57-79]. In this paper, the chemical trees with the maximum first/secon…
A classical result of Komlós, Sárközy and Szemerédi states that every $n$-vertex graph with minimum degree at least $(1/2+ o(1))n$ contains every $n$-vertex tree with maximum degree $O(n/\log{n})$ as a subgraph, and the bounds on the degree conditions are sharp. On the other hand, Krivelevich, Kwan and Sudakov recently proved that for every $n$-vertex graph $G_α$ with minimum degree at least $αn$ for any fixed $α>0$ and every $n$-vertex tree $T$ with bounded maximum degree, one can still find a copy of $T$ in $G_α$ with high probability after adding $O(n)$ randomly-chosen edges to $G_α$. We ex…
The Wiener index is maximized over the set of trees with the given vertex weight and degree sequences. This model covers the traditional "unweighed" Wiener index, the terminal Wiener index, and the vertex distance index. It is shown that there exists an optimal caterpillar. If weights of internal vertices increase in their degrees, then an optimal caterpillar exists with weights of internal vertices on its backbone monotonously increasing from some central point to the ends of the backbone, and the same is true for pendent vertices. A tight upper bound of the Wiener index value is proposed and…
Algorithm for finding vertex-edge domination number on graphs with bounded treewidth and related problems on planar graphs
本文提出在界树宽图中求解最小顶点-边支配集的多项式时间算法，并证明平面图的树宽与顶点-边支配数的关系。
The $k$-rainbow index $rx_k(G)$ of a connected graph $G$ was introduced by Chartrand, Okamoto and Zhang in 2010. As a natural counterpart of the $k$-rainbow index, we introduced the concept of $k$-vertex-rainbow index $rvx_k(G)$ in this paper. For a graph $G=(V,E)$ and a set $S\subseteq V$ of at least two vertices, \emph{an $S$-Steiner tree} or \emph{a Steiner tree connecting $S$} (or simply, \emph{an $S$-tree}) is a such subgraph $T=(V',E')$ of $G$ that is a tree with $S\subseteq V'$. For $S\subseteq V(G)$ and $|S|\geq 2$, an $S$-Steiner tree $T$ is said to be a \emph{vertex-rainbow $S$-tree}…
Embedding loose spanning trees in 3-uniform hypergraphs
本文将Komlós–Sárközy–Szemerédi关于图中生成树嵌入的经典结果推广到3-均匀超图，证明最小顶点度达到$(5/9+γ)\inom{n}{2}$时必含任意有界最大度的松散生成树，并说明该界渐近最优。
The eccentricity of a vertex $u$ in a connected graph $G$ is the distance between $u$ and a vertex farthest from it; the eccentric sequence of $G$ is the nondecreasing sequence of the eccentricities of $G$. In this paper, we determine the unique tree that minimises the Wiener index, i.e. the sum of distances between all unordered vertex pairs, among all trees with a given eccentric sequence. We show that the same tree maximises the number of subtrees among all trees with a given eccentric sequence, thus providing another example of negative correlation between the number of subtrees and the Wi…
We calculate the exact number of contours of size $n$ containing a fixed vertex in $d$-ary trees and provide sharp estimates for this number for more general trees. We also obtain a characterization of the locally finite trees with infinitely many contours of the same size containing a fixed vertex.
Induced subgraphs and tree decompositions XI. Local structure in even-hole-free graphs of large treewidth
本文证明Sintiari-Trotignon猜想：大树宽无偶环图必含四顶点五边诱导子图，并推广到更一般的弦图与森林情形，且证明选择最优。
In a rooted tree, we call a vertex {\em balanced} if it is at equal distance from all its descendant leaves. We count balanced vertices in three different tree varieties. For decreasing binary trees, we can prove that the probability that a vertex chosen uniformly at random from the set of all trees of a given size is balanced is monotone decreasing.
The vertex sets of subtrees of a tree
本文研究集合族能否表示为某棵树的所有子树的顶点集，证明在有限情形及一定无限条件下，Helly性质与交图弦性这两个必要条件也是充分的。
Topological Indices Among Strong Support Vertex
研究具有强支撑顶点的树的Albertson指数与Sigma指数，分析其不规则性并建立给定度序列下的极值界。
On the minimum number of maximal distance-$k$ independent sets in trees
研究确定了n顶点树中最大距离-k独立集的最小可能数量，给出了公式并分析了达到此界限的树类及数量增长速率。
On Universal Graphs for Trees and Tree-Like Graphs
本文改进了陈和格雷厄姆关于树的通用图边数界，并推广至树宽为k的图。
A $(q,r)$\emph{-tree-coloring} of a graph $G$ is a $q$-coloring of vertices of $G$ such that the subgraph induced by each color class is a forest of maximum degree at most $r.$ An \emph{equitable $(q, r)$-tree-coloring} of a graph $G$ is a $(q,r)$-tree-coloring such that the sizes of any two color classes differ by at most one. Let the \emph{strong equitable vertex $r$-arboricity} be the minimum $p$ such that $G$ has an equitable $(q, r)$-tree-coloring for every $q\geq p.$ In this paper, we find the exact value for each $va^\equiv_2(K_{m,n})$ and $va^\equiv_2(K_{l,m,n}).$
The eccentricity of a vertex $v$ in a graph $G$ is the maximum distance from $v$ to any other vertex. The vertices whose eccentricity are equal to the diameter (the maximum eccentricity) of $G$ are called peripheral vertices. In trees the eccentricity at $v$ can always be achieved by the distance from $v$ to a peripheral vertex. From this observation we are motivated to introduce normality of a vertex $v$ as the minimum distance from $v$ to any peripheral vertex. We consider the properties of the normality as well as the middle part of a tree with respect to this concept. Various related obser…
We provide a new derivation of the well-known generating function counting the number of walks on a regular tree that start and end at the same vertex, and more generally, a generating function for the number of walks that end at a vertex a distance i from the start vertex. These formulas seem to be very old, and go back, in an equivalent form, at least to Harry Kesten's work on symmetric random walks on groups from 1959, and in the present form to Brendan McKay (1983).
Let $T$ be a tree. A vertex of degree one is a \emph{leaf} of $T$ and a vertex of degree at least three is a \emph{branch vertex} of $T$. A graph is said to be \emph{$K_{1,4}$-free} if it does not contain $K_{1,4}$ as an induced subgraph. In this paper, we study the spanning trees with a bounded number of leaves and branch vertices of $K_ {1,4}$-free graphs. Applying the main results, we also give some improvements of previous results on the spanning tree with few branch vertices for the case of $K_{1,4}$-free graphs.
Trees with many leaves in tournaments
本文研究竞赛图中包含有向树的条件，基于叶子数量和最大度数改进了Sumner通用竞赛图猜想，证明了Havet和Thomassé猜想的渐近形式，并改进了Mycroft和Naia关于最大度数限制的结果。
In this paper we find recurrence relations for the asymptotic probability a vertex is $k$ protected in all Motzkin trees. We use a similar technique to calculate the probabilities for balanced vertices of rank $k$. From this we calculate upper and lower bounds for the probability a vertex is balanced and upper and lower bounds for the expected rank of balanced vertices.
A vertex of degree one is called an end-vertex, and an end-vertex of a tree is called a leaf. A tree with at most $k$ leaves is called a $k$-ended tree. For a positive integer $k$, let $t_k$ be the order of a largest $k$-ended tree. Let $σ_m$ be the minimum degree sum of an independent set of $m$ vertices. The main result (Theorem 2) provides a lower bound for $t_{k+1}$ in terms of $σ_m$ and relative orders: if $G$ is a connected graph and $k$, $λ$, $m$ are positive integers with $2\le m\le\min\{k,λ\}+1$ then either $t_{k+1} \ge σ_m +λ(k-m+1)+1$ or $t_k\ge t_{k+1}-λ+1$.
Let $T$ be a tree, a vertex of degree one and a vertex of degree at least three is called a leaf and a branch vertex, respectively. The set of leaves of $T$ is denoted by $Leaf(T)$. The subtree $T-Leaf(T)$ of $T$ is called the stem of $T$ and denoted by $Stem(T).$ In this paper, we give two sufficient conditions for a connected claw-free graph to have a spanning tree whose stem has a bounded number of branch vertices, and those conditions are best possible. As corollaries of main results we also give some conditions to show that a connected claw-free graph has a spanning tree whose stem is a s…
Let $G =(V_{G}, E_{G})$ be a simple connected graph with its vertex set $V_{G}$ and edge set $E_{G}$. The Mostar index $Mo(G)$ was defined as $Mo(G)=\sum\limits_{e=uv\in E(G)}|n_{u}-n_{v}|$, where $n_{u}$ (resp., $n_{v}$) is the number of vertices whose distance to vertex $u$ (resp., $v$) is smaller than the distance to vertex $v$ (resp., $u$). In this study, we determine the first three minimum Mostar indices of tree-like phenylenes and characterize all the tree-like phenylenes attaining these values. At last, we give some numerical examples and discussion.
Cayley trees and increasing 1,2-trees: let's twist!
本文证明递增1,2-树与Cayley树计数相同，其中三角形数对应扭转数，并给出三个证明及高效随机生成器。
We characterize the extremal structures for mixing walks on trees that start from the most advantageous vertex. Let $G=(V,E)$ be a tree with stationary distribution $π$. For a vertex $v \in V$, let $H(v,π)$ denote the expected length of an optimal stopping rule from $v$ to $π$. The \emph{best mixing time} for $G$ is $\min_{v \in V} H(v,π)$. We show that among all trees with $|V|=n$, the best mixing time is minimized uniquely by the star. For even $n$, the best mixing time is maximized by the uniquely path. Surprising, for odd $n$, the best mixing time is maximized uniquely by a path of length …
In this paper, we continue the study of the total domination game in graphs introduced in [Graphs Combin. 31(5) (2015), 1453--1462], where the players Dominator and Staller alternately select vertices of $G$. Each vertex chosen must strictly increase the number of vertices totally dominated, where a vertex totally dominates another vertex if they are neighbors. This process eventually produces a total dominating set $S$ of $G$ in which every vertex is totally dominated by a vertex in $S$. Dominator wishes to minimize the number of vertices chosen, while Staller wishes to maximize it. The game …
On local antimagic vertex coloring for complete full $t$-ary trees
本文验证了关于完全满 $t$ 叉树的局部反魔幻色数猜想，并证明当 $t$ 为奇数时，其精确值为叶子数加一。
We investigate the problem of ordering trees according to their Laplacian energy. More precisely, given a positive integer $n$, we find a class of cardinality approximately $\sqrt{n}$ whose elements are the $n$-vertex trees with largest Laplacian energy. The main tool for establishing this result is a new upper bound on the sum $S_k(T)$ of the $k$ largest Laplacian eigenvalues of an $n$-vertex tree $T$ with diameter at least four, where $k \in \{1,...,n\}$.
Broadcast independence and packing in certain classes of trees
本文研究完美 k 叉树、蜘蛛图和双蜘蛛图中最大权重的独立广播与 packing 广播问题，部分回答了 Ahmane 等人提出的问题。
We use a sign-reversing involution to show that trees on the vertex set [n], considered to be rooted at 1, in which no vertex has exactly one child are counted by 1/n sum_{k=1}^{n} (-1)^(n-k) {n}-choose-{k} (n-1)!/(k-1)! k^(k-1). This result corrects a persistent misprint in the Encyclopedia of Integer Sequences.
Sustainable trees
本研究针对树的边可持续性展开，定义了可删除替换后保持连通且谱半径不降低的边为可持续边，探究其基本性质，证明含大度数顶点的树的相关结论，分类边界情形下的可持续树并提出开放问题。
A recent result of Condon, Kim, Kühn and Osthus implies that for any $r\geq (\frac{1}{2}+o(1))n$, an $n$-vertex almost $r$-regular graph $G$ has an approximate decomposition into any collections of $n$-vertex bounded degree trees. In this paper, we prove that a similar result holds for an almost $αn$-regular graph $G$ with any $α>0$ and a collection of bounded degree trees on at most $(1-o(1))n$ vertices if $G$ does not contain large bipartite holes. This result is sharp in the sense that it is necessary to exclude large bipartite holes and we cannot hope for an approximate decomposition into …
Given a collection of graphs $\mathbf{G}=(G_1, \ldots, G_m)$ with the same vertex set, an $m$-edge graph $H\subset \cup_{i\in [m]}G_i$ is a transversal if there is a bijection $ϕ:E(H)\to [m]$ such that $e\in E(G_{ϕ(e)})$ for each $e\in E(H)$. We give asymptotically-tight minimum degree conditions for a graph collection on an $n$-vertex set to have a transversal which is a copy of a graph $H$, when $H$ is an $n$-vertex graph which is an $F$-factor or a tree with maximum degree $o(n/\log n)$.
Spanning trees of $K_{1,4}$-free graphs whose reducible stems have few leaves
本文研究 $K_{1,4}$-自由图，给出其存在可约干叶子数较少的生成树的尖锐充分条件。
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