Let $k$ and $n$ be integers such that $1\leq k \leq n-1$, and let $G$ be a simple graph of order $n$. The $k$-token graph $F_k(G)$ of $G$ is the graph whose vertices are the $k$-subsets of $V(G)$, where two vertices are adjacent in $F_k(G)$ whenever their symmetric difference is an edge of $G$. In this paper we show that if $G$ is a tree, then the connectivity of $F_k(G)$ is equal to the minimum degree of $F_k(G)$.
We study the algebraic connectivity (or second Laplacian eigenvalue) of token graphs, also called symmetric powers of graphs. The $k$-token graph $F_k(G)$ of a graph $G$ is the graph whose vertices are the $k$-subsets of vertices from $G$, two of which being adjacent whenever their symmetric difference is a pair of adjacent vertices in $G$. Recently, it was conjectured that the algebraic connectivity of $F_k(G)$ equals the algebraic connectivity of $G$. In this paper, we prove the conjecture for new infinite families of graphs, such as trees and graphs with maximum degree large enough.
The concept of pendant-tree connectivity, introduced by Hager in 1985, is a generalization of classical vertex-connectivity. In this paper, we study pendant-tree connectivity of line graphs.
In this paper we establish all extremal graphs with respect to augmented eccentric connectivity index among all (simple connected) graphs, among trees and among trees with perfect matching. For graphs that turn out to be extremal explicit formulas for the value of augmented eccentric connectivity index are derived.
Random increasing k-trees represent an interesting, useful class of strongly dependent graphs for which analytic-combinatorial tools can be successfully applied. We study in this paper a notion called connectivity-profile and derive asymptotic estimates for it; some interesting consequences will also be given.
We obtain the maximum sum-connectivity indices of graphs in the set of trees and in the set of unicyclic graphs respectively with given number of vertices and maximum degree, and determine the corresponding extremal graphs. Additionally, we deduce the n-vertex unicyclic graphs with the first two maximum sum-connectivity indices for $n\ge 4$.
The concept of pedant tree-connectivity was introduced by Hager in 1985. For a graph $G=(V,E)$ and a set $S\subseteq V(G)$ 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 an $S$-Steiner tree, if the degree of each vertex in $S$ is equal to one, then this tree is called a \emph{pedant $S$-Steiner tree}. Two pedant $S$-Steiner trees $T$ and $T'$ are said to be \emph{internally disjoint} if $E(T)\cap E(T')=\varnothing$ and $V(T)\cap V(T')=S$. …
For a digraph $D=(V(D), A(D))$, and a set $S\subseteq V(D)$ with $r\in S$ and $|S|\geq 2$, an $(S, r)$-tree is an out-tree $T$ rooted at $r$ with $S\subseteq V(T)$. Two $(S, r)$-trees $T_1$ and $T_2$ are said to be arc-disjoint if $A(T_1)\cap A(T_2)=\emptyset$. Two arc-disjoint $(S, r)$-trees $T_1$ and $T_2$ are said to be internally disjoint if $V(T_1)\cap V(T_2)=S$. Let $κ_{S,r}(D)$ and $λ_{S,r}(D)$ be the maximum number of internally disjoint and arc-disjoint $(S, r)$-trees in $D$, respectively. The generalized $k$-vertex-strong connectivity of $D$ is defined as $$κ_k(D)= \min \{κ_{S,r}(D)\…
The eccentric connectivity index $ξ^c$ is a distance--based molecular structure descriptor that was recently used for mathematical modeling of biological activities of diverse nature. We prove that the broom has maximum $ξ^c$ among trees with a fixed maximum vertex degree, and characterize such trees with minimum $ξ^c$\,. In addition, we propose a simple linear algorithm for calculating $ξ^c$ of trees.
The Steiner tree problem in graphs has applications in network design or circuit layout. Given a set $S$ of vertices, $|S| \geq 2,$ a tree connecting all vertices of $S$ is called an $S$-Steiner tree (tree connecting $S$). The reliability of a network $G$ to connect any $S$ vertices ($|S|$ number of vertices) in $G$ can be measure by this parameter. For an $S$-Steiner tree, if the degree of each vertex in $S$ is equal to one, then that tree is called a pendant S-Steiner tree. Two pendant $S$-Steiner trees $T$ and $T'$ are said to be internally disjoint if $E(T) \cap E(T') = \emptyset$ and $V(T…
A $k$-ended tree is a tree with at most $k$ leaves. In this note, we give a simple proof for the following theorem. Let $G$ be a connected graph and $k$ be an integer ($k\geq 2$). Let $S$ be a vertex subset of $G$ such that $α_{G}(S) \leq k + κ_{G}(S)- 1.$ Then, $G$ has a $k$-ended tree which covers $S.$ Moreover, the condition is sharp.
Let $G$ be a simple graph of order $n\geq 2$ and let $k\in \{1,\ldots ,n-1\}$. The $k$-token graph $F_k(G)$ of $G$ is the graph whose vertices are the $k$-subsets of $V(G)$, where two vertices are adjacent in $F_k(G)$ whenever their symmetric difference is an edge of $G$. In 2018 J. Leaños and A. L. Trujillo-Negrete proved that if $G$ is $t$-connected and $t\geq k$, then $F_k(G)$ is at least $k(t-k+1)$-connected. In this paper we show that such a lower bound remains true in the context of edge-connectivity. Specifically, we show that if $G$ is $t$-edge-connected and $t\geq k$, then $F_k(G)$ is…
Hasunuma [J. Graph Theory 102 (2023) 423-435] conjectured that for any tree $T$ of order $m$, every $k$-connected (or $k$-edge-connected) graph $G$ with minimum degree at least $k+m-1$ contains a tree $T'\cong T$ such that $G-E(T')$ is still $k$-connected (or $k$-edge connected). Hasunuma verified this conjecture for $k\leq 2$. In this paper, we confirm this conjecture for $k=3$.
The concept of pendant-tree $k$-connectivity $τ_k(G)$ of a graph $G$, introduced by Hager in 1985, is a generalization of classical vertex-connectivity. Let $f(n,k,\ell)$ be the minimal number of edges of a graph $G$ of order $n$ with $τ_k(G)=\ell \ (1\leq \ell\leq n-k)$. In this paper, we give some exact value or sharp bounds of the parameter $f(n,k,\ell)$.
The Algebraic Connectivity and Laplacian Spectral Radius of Token Graphs
Shenzhen University of Information Technology（深圳信息职业技术学院）；Universitat de Lleida（莱里达大学）；Universitat Politècnica de Catalunya（加泰罗尼亚理工大学）；Northwestern Polytechnical University（西北工业大学）
For a graph $G=(V,E)$ and a set $S\subseteq V(G)$ of size at least $2$, an $S$-Steiner tree $T$ is a subgraph of $G$ that is a tree with $S\subseteq V(T)$. Two $S$-Steiner trees $T$ and $T'$ are internally disjoint (resp. edge-disjoint) if $E(T)\cap E(T')=\emptyset$ and $V(T)\cap V(T')=S$ (resp. if $E(T)\cap E(T')=\emptyset$). Let $κ_G (S)$ (resp. $λ_G (S)$) denote the maximum number of internally disjoint (resp. edge-disjoint) $S$-Steiner trees in $G$. The $k$-tree connectivity $κ_k(G)$ (resp. $k$-tree edge-connectivity $λ_k(G)$) of $G$ is then defined as the minimum $κ_G (S)$ (resp. $λ_G (S)…
The recent paper "Linear Connectivity Forces Large Complete Bipartite Minors" by Boehme et al. relies on a structure theorem for graphs with no H-minor. The sketch provided of how to deduce this theorem from the work of Robertson and Seymour appears to be incomplete. To fill this gap, we modify the main proof of that paper to work with a mere restatement of Robertson and Seymour's original results instead.
The general sum-connectivity index of a graph $G$ is defined as $χ_α(G)=\sum\limits_{uv\in E(G)} {(d(u)+d(v))^α}$, where $d(v)$ denotes the degree of the vertex $v$ in $G$ and $α$ is a real number. In this paper it is deduced the maximum value for the general sum-connectivity index of $n$-vertex trees for $-1.7036\leq α<0$ and of $n$-vertex unicyclic graphs for $-1\le α<0$ respectively, with fixed maximum degree $\triangle $. The corresponding extremal graphs, as well as the $n$-vertex unicyclic graphs with the second maximum general sum-connectivity index for $n\ge 4$ are characterized. This …
The recently developed atom-bond sum-connection (ABS) index is a variation of the well-studied graph-based molecular descriptors connectivity (Randic), atom-bond connectivity, and sum-connectivity indices.We present the minimum ABS index of trees with a given number of pendent vertices in this paper, which provides a solution to the problem proposed by A. Ali, I. Gutman and I. Redzepovic, Atom-Bond Sum-Connectivity Index of Unicyclic Graphs and Some Applications, Electron. J. Math.(2023) 1-7
For a digraph $D=(V(D), A(D))$, and a set $S\subseteq V(D)$ with $r\in S$ and $|S|\geq 2$, an $(S, r)$-tree is an out-tree $T$ rooted at $r$ with $S\subseteq V(T)$. Two $(S, r)$-trees $T_1$ and $T_2$ are said to be arc-disjoint if $A(T_1)\cap A(T_2)=\emptyset$. Two arc-disjoint $(S, r)$-trees $T_1$ and $T_2$ are said to be internally disjoint if $V(T_1)\cap V(T_2)=S$. Let $κ_{S,r}(D)$ and $λ_{S,r}(D)$ be the maximum number of internally disjoint and arc-disjoint $(S, r)$-trees in $D$, respectively. The generalized $k$-vertex-strong connectivity of $D$ is defined as $$κ_k(D)= \min \{κ_{S,r}(D)\…
The eccentric connectivity index, proposed by Sharma, Goswami and Madan, has been employed successfully for the development of numerous mathematical models for the prediction of biological activities of diverse nature. We now report mathematical properties of the eccentric connectivity index. We establish various lower and upper bounds for the eccentric connectivity index in terms of other graph invariants including the number of vertices, the number of edges, the degree distance and the first Zagreb index. We determine the n-vertex trees of diameter with the minimum eccentric connectivity ind…
We investigate the structure of trees that have minimal algebraic connectivity among all trees with a given degree sequence. We show that such trees are caterpillars and that the vertex degrees are non-decreasing on every path on non-pendant vertices starting at the characteristic set of the Fiedler vector.
We show that every graph admits a canonical tree-like decomposition into its $k$-edge-connected pieces for all $k\in\mathbb{N}\cup\{\infty\}$ simultaneously.
In this paper, we present some new results describing connections between the spectrum of a regular graph and its generalized connectivity, toughness, and the existence of spanning trees with bounded degree.
Given a graph or a matroid, a tree of tangles is a tree decomposition that displays the structure of the connectivity: every edge of the decomposition tree induces a separation, that is, a way to divide the graph or matroid into two parts; and for every two highly connected areas (encoded as tangles) that live on different sides of some separation, some separation induced by an edge distinguishes them. Separations induced by a tree of tangles cannot cross. One approach to display even more connectivity structure is to insert even more structure into a tree of tangles, for example the flowers t…
Let $\mathcal{G}$ be the set of simple graphs (or multigraphs) $G$ such that for each $G \in \mathcal{G}$ there exists at least two non-empty disjoint proper subsets $V_{1},V_{2}\subseteq V(G)$ satisfying $V(G)\setminus(V_{1} \cup V_{2})\neq ϕ$ and edge connectivity $κ'(G)=e(V_{i},V(G)\backslash V_{i})$ for $1\leq i \leq 2$. A multigraph is a graph with possible multiple edges, but no loops. Let $τ(G)$ be the maximum number of edge-disjoint spanning trees of a graph $G$. Motivated by a question of Seymour on the relationship between eigenvalues of a graph $G$ and bounds of $τ(G)$, we mainly gi…
Considering systems of separations in a graph that separate every pair of a given set of vertex sets that are themselves not separated by these separations, we determine conditions under which such a separation system contains a nested subsystem that still separates those sets and is invariant under the automorphisms of the graph. As an application, we show that the $k$-blocks -- the maximal vertex sets that cannot be separated by at most $k$ vertices -- of a graph $G$ live in distinct parts of a suitable tree-decomposition of $G$ of adhesion at most $k$, whose decomposition tree is invariant …
The {\em atom-bond connectivity (ABC) index} is a degree-based molecular descriptor, that found chemical applications. It is well known that among all connected graphs, the graphs with minimal ABC index are trees. A complete characterization of trees with minimal $ABC$ index is still an open problem. In this paper, we present new structural properties of trees with minimal ABC index. Our main results reveal that trees with minimal ABC index do not contain so-called {\em $B_k$-branches}, with $k \geq 5$, and that they do not have more than four $B_4$-branches.
For a graph $G = (V, E)$, a subset $F\subset V(G)$ is called an $R_k$-vertex-cut of $G$ if $G -F$ is disconnected and each vertex $u \in V(G)- F$ has at least $k$ neighbors in $G -F$. The $R_k$-vertex-connectivity of $G$, denoted by $κ^k(G)$, is the cardinality of the minimum $R_k$-vertex-cut of $G$, which is a refined measure for the fault tolerance of network $G$. In this paper, we study $κ^2$ for Cayley graphs generated by $k$-trees. Let $Sym(n)$ be the symmetric group on $\{1, 2, \cdots ,n\}$ and $\mathcal{T}$ be a set of transpositions of $Sym(n)$. Let $G(\mathcal{T})$ be the graph on $n$…
We consider $2$-colourings $f : E(G) \rightarrow \{ -1 ,1 \}$ of the edges of a graph $G$ with colours $-1$ and $1$ in $\mathbb{Z}$. A subgraph $H$ of $G$ is said to be a zero-sum subgraph of $G$ under $f$ if $f(H) := \sum_{e\in E(H)} f(e) =0$. We study the following type of questions, in several cases obtaining best possible results: Under which conditions on $|f(G)|$ can we guarantee the existence of a zero-sum spanning tree of $G$? The types of $G$ we consider are complete graphs, $K_3$-free graphs, $d$-trees, and maximal planar graphs. We also answer the question of when any such colouring…
The concept of maximum local connectivity $\bar κ$ of a graph was introduced by Bollobás. One of the problems about it is to determine the largest number of edges $f(n;\barκ\leq \ell)$ for graphs of order $n$ that have local connectivity at most $\ell$. We consider a generalization of the above concept and problem. For $S\subseteq V(G)$ and $|S|\geq 2$, the \emph{generalized local connectivity} $κ(S)$ is the maximum number of internally disjoint trees connecting $S$ in $G$. The parameter $\barκ_k(G)=max\{κ(S)|S\subseteq V(G),|S|=k\}$ is called the \emph{maximum generalized local connectivity} …
With graphs considered as natural models for many network design problems, edge connectivity $κ'(G)$ and maximum number of edge-disjoint spanning trees $τ(G)$ of a graph $G$ have been used as measures for reliability and strength in communication networks modeled as graph $G$ (see \cite{Cunn85, Matula87}, among others). Mader \cite{Mader71} and Matula \cite{Matula72} introduced the maximum subgraph edge connectivity $\overline{κ'}(G)=\max \{κ'(H): H \mbox{ is a subgraph of } G \}$. Motivated by their applications in network design and by the established inequalities \[ \overline{κ'}(G)\ge κ'(G…
Let $τ(G)$ and $κ'(G)$ denote the edge-connectivity and the spanning tree packing number of a graph $G$, respectively. Proving a conjecture initiated by Cioaba and Wong, Liu et al. in 2014 showed that for any simple graph $G$ with minimum degree $δ\ge 2k \ge 4$, if the second largest adjacency eigenvalue of $G$ satisfies $λ_2(G) < δ- \frac{2k-1}{δ+1}$, then $τ(G) \ge k$. Similar results involving the Laplacian eigenvalues and the signless Laplacian eigenvalues of $G$ are also obtained. In this paper, we find a function $f(δ, k, g)$ such that for every graph $G$ with minimum degree $δ\ge 2k \ge…
Mader conjectured that for any tree $T$ of order $m$, every $k$-connected graph $G$ with minimum degree at least $\lfloor\frac{3k}{2}\rfloor +m-1$ contains a subtree $T'\cong T$ such that $G-V(T')$ is $k$-connected. In this paper, we give a characterization for a subgraph to contain an embedding of a specified tree avoiding some vertex. As a corollary, we confirm Mader's conjecture for $k\leq3$.
The {\em atom-bond connectivity (ABC) index} is a degree-based graph topological index that found chemical applications. The problem of complete characterization of trees with minimal $ABC$ index is still an open problem. In~\cite{d-sptmabci-2014}, it was shown that trees with minimal ABC index do not contain so-called {\em $B_k$-branches}, with $k \geq 5$, and that they do not have more than four $B_4$-branches. Our main results here reveal that the number of $B_1$ and $B_2$-branches are also bounded from above by small fixed constants. Namely, we show that trees with minimal ABC index do not…
Let $S$ be a nonempty set of vertices of a connected graph $G$. A collection $T_1,..., T_\ell$ of trees in $G$ is said to be internally disjoint trees connecting $S$ if $E(T_i)\cap E(T_j)= \emptyset$ and $V(T_i)\cap V(T_j)=S$ for any pair of distinct integers $i, j$, where $1 \leq i, j \leq r$. For an integer $k$ with $2 \leq k \leq n$, the generalized $k$-connectivity $κ_k(G)$ of $G$ is the greatest positive integer $r$ such that $G$ contains at least $r$ internally disjoint trees connecting $S$ for any set $S$ of $k$ vertices of $G$. Obviously, $κ_2(G)$ is the connectivity of $G$. In this pa…
The concept of pedant tree-connectivity was introduced by Hager in 1985. For a graph $G=(V,E)$ and a set $S\subseteq V(G)$ 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 an $S$-Steiner tree, if the degree of each vertex in $S$ is equal to one, then this tree is called a \emph{pedant $S$-Steiner tree}. Two pedant $S$-Steiner trees $T$ and $T'$ are said to be \emph{internally disjoint} if $E(T)\cap E(T')=\varnothing$ and $V(T)\cap V(T')=S$. …
The {\em atom-bond connectivity (ABC) index} is one of the recently most investigated degree-based molecular structure descriptors, that have applications in chemistry. For a graph $G$, the ABC index is defined as $\sum_{uv\in E(G)}\sqrt{\frac{(d(u) +d(v)-2)}{d(u)d(v)}}$, where $d(u)$ is the degree of vertex $u$ in $G$ and $E(G)$ is the set of edges of $G$. Despite many attempts in the last few years, it is still an open problem to characterize trees with minimal $ABC$ index. In this paper, we present an efficient approach of computing trees with minimal ABC index, by considering the degree se…
A \emph{vertex separator} of a connected graph $G$ is a set of vertices removing which will result in two or more connected components and a \emph{minimum vertex separator} is a set which contains the minimum number of such vertices, i.e., the cardinality of this set is least among all possible vertex separator sets. The cardinality of the minimum vertex separator refers to the connectivity of the graph G. A connected graph is said to be $k-connected$ if removing exactly $k$ vertices, $ k\geq 1$, from the graph, will result in two or more connected components and on removing any $(k-1)$ vertic…
The edge-connectivity of a graph is the minimum number of edges whose deletion disconnects the graph. Let $Δ(G)$ the maximum degree of a graph $G$ and let $ρ(G)$ be the spectral radius of $G$. In this article we present a lower bound for $Δ(G)-ρ(G)$ in terms of the edge connectivity of $G$, where $G$ is a nonregular distance-hereditary graph. We also prove that $ρ(G)$ reaches the maximum at a unique graph in $\mathcal G$, when $\vert V(G)\vert = n$, and $\mathcal G$ either is in the class of graphs with bounded tree-width or is in the class of block graphs with prescribed independence number.
For a connected graph, a {\em minimum vertex separator} is a minimum set of vertices whose removal creates at least two connected components. The vertex connectivity of the graph refers to the size of the minimum vertex separator and a graph is $k$-vertex connected if its vertex connectivity is $k$, $k\geq 1$. Given a $k$-vertex connected graph $G$, the combinatorial problem {\em vertex connectivity augmentation} asks for a minimum number of edges whose augmentation to $G$ makes the resulting graph $(k+1)$-vertex connected. In this paper, we initiate the study of $r$-vertex connectivity augmen…
The generalized connectivity of a graph, which was introduced recently by Chartrand et al., is a generalization of the concept of vertex connectivity. Let $S$ be a nonempty set of vertices of $G$, a collection $\{T_1,T_2,...,T_r\}$ of trees in $G$ is said to be internally disjoint trees connecting $S$ if $E(T_i)\cap E(T_j)=\emptyset$ and $V(T_i)\cap V(T_j)=S$ for any pair of distinct integers $i,j$, where $1\leq i,j\leq r$. For an integer $k$ with $2\leq k\leq n$, the $k$-connectivity $κ_k(G)$ of $G$ is the greatest positive integer $r$ for which $G$ contains at least $r$ internally disjoint t…
Let $G$ be a nontrivial connected graph of order $n$, and $k$ an integer with $2\leq k\leq n$. For a set $S$ of $k$ vertices of $G$, let $κ(S)$ denote the maximum number $\ell$ of edge-disjoint trees $T_1,T_2,...,T_\ell$ in $G$ such that $V(T_i)\cap V(T_j)=S$ for every pair $i,j$ of distinct integers with $1\leq i,j\leq \ell$. Chartrand et al. generalized the concept of connectivity as follows: The $k$-$connectivity$, denoted by $κ_k(G)$, of $G$ is defined by $κ_k(G)=$min$\{κ(S)\}$, where the minimum is taken over all $k$-subsets $S$ of $V(G)$. Thus $κ_2(G)=κ(G)$, where $κ(G)$ is the connectiv…
A conjecture of Luo, Tian and Wu (2022) says that for every positive integer $k$ and every finite tree $T$ with bipartition $X$ and $Y$ (denote $t = \max\{|X|,|Y |\})$, every $k$-connected bipartite graph $G$ with $δ(G) \geq k + t$ contains a subtree $T' \cong T$ such that $κ(G-V (T')) \geq k$. In this paper, we confirm this conjecture for caterpillars when $k=3$ and spiders when $k\leq 3$.
Connectivity keeping trees in triangle-free graphs
Korea Institute for Advanced Study（韩国高等科学研究院）；School of Data Science, Yokohama City University（横滨市立大学数据科学学院）；Department of Mathematics Education, Seoul National University（首尔国立大学数学教育系）；Department of Mathematics, Ajou University（阿丘大学数学系）
The eccentric connectivity index $ξ^c$ is a novel distance--based molecular structure descriptor that was recently used for mathematical modeling of biological activities of diverse nature. It is defined as $ξ^c (G) = \sum_{v \in V (G)} deg (v) \cdot ε(v)$\,, where $deg (v)$ and $ε(v)$ denote the vertex degree and eccentricity of $v$\,, respectively. We survey some mathematical properties of this index and furthermore support the use of eccentric connectivity index as topological structure descriptor. We present the extremal trees and unicyclic graphs with maximum and minimum eccentric connect…
