Arbind K. Lal, Kamal L. Patra, Binod K. Sahoo
In this paper we consider the following problem: Over the class of all simple connected graphs of order $n$ with $k$ pendant vertices ($n,k$ being fixed), which graph maximizes (respectively, minimizes) the algebraic connectivity? We also discuss the algebraic connectivity of unicyclic graphs.
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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.
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…
For a graph $G$, the first multiplicative Zagreb index $\prod_1(G) $ is the product of squares of vertex degrees, and the second multiplicative Zagreb index $\prod_2(G) $ is the product of products of degrees of pairs of adjacent vertices. In this paper, we explore graphs with extremal $Π_{1}(G)$ and $Π_{2}(G)$ in terms of (edge) connectivity and pendant vertices. The corresponding extremal graphs are characterized with given connectivity at most $k$ and $p$ pendant vertices. In addition, the maximum and minimum values of $\prod_1(G) $ and $\prod_2(G) $ are provided. Our results extend and enr…
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 Wiener index of a connected graph is defined as the sum of the distances between all unordered pair of its vertices. In this paper, we characterize the graphs which extremize the Wiener index among all graphs on $n$ vertices with $k$ pendant vertices. We also characterize the graph which minimizes the Wiener index over the graphs on $n$ vertices with $s$ cut vertices.
The Sombor index is a topological index in graph theory defined by Gutman in 2021. In this article we find the maximum Sombor index of unicyclic graphs with a fixed number of pendant vertices. We also provide the unique graph among the chosen class where the maximum Sombor index is attained.
In this paper we determine the graph whose least eigenvalue of signless Laplacian attains the minimum or maximum among all connected non-bipartite graphs of fixed order and given number of pendant vertices. Thus we obtain a lower bound and an upper bound for the least eigenvalue of signless Laplacian of a graph in terms of the number of pendent vertices.
One theorem of Nemhauser and Trotter ensures that, under certain conditions, a stable set of a graph G can be enlarged to a maximum stable set of this graph. For example, any stable set consisting of only simplicial vertices is contained in a maximum stable set of G. In this paper we demonstrate that an inverse assertion is true for trees of order greater than one, where, in fact, all the simplicial vertices are pendant. Namely, we show that any maximum stable set of such a tree contains at least one pendant vertex. Moreover, we prove that if T does not own a perfect matching, then a stable se…
The total eccentricity index of a connected graph is defined as sum of the eccentricities of all its vertices. We denote the set of all connected graphs on $n$ vertices with $k$ pendant vertices by $\mathfrak{H}_{n,k}$ and denote the set of all connected graphs on $n$ vertices with $s$ cut vertices by $\mathfrak{C_{n,s}}$. In this paper, we give the sharp lower and upper bounds on the total eccentricity index over $\mathfrak{H}_{n,k}$ and the sharp lower bound for the same over $\mathfrak{C_{n,s}}$. We also provide the sharp upper bounds on the total eccentricity index over $\mathfrak{C_{n,s}}…
A vertex whose removal in a graph $G$ increases the number of components of $G$ is called a cut vertex. For all $n,c$, we determine the maximum number of connected induced subgraphs in a connected graph with order $n$ and $c$ cut vertices, and also characterise those graphs attaining the bound. Moreover, we show that the cycle has the smallest number of connected induced subgraphs among all cut vertex-free connected graphs. The general case $c>0$ remains an open task. We also characterise the extremal graph structures given both order and number of pendant vertices, and establish the correspon…
A tree is called double starlike if it has exactly two vertices of degree greater than two. Let $H(p,n,q)$ denote the double starlike tree obtained by attaching $p$ pendant vertices to one pendant vertex of the path $P_n$ and $q$ pendant vertices to the other pendant vertex of $P_n$. In this paper, we prove that $H(p,n,q)$ is determined by its Laplacian spectrum.
A stable set in a graph G is a set of mutually non-adjacent vertices, alpha(G) is the size of a maximum stable set of G, and core(G) is the intersection of all its maximum stable sets. In this paper we demonstrate that in a tree T, of order n greater than 1, any stable set of size greater or equal to n/2 contains at least one pendant vertex. Hence, we deduce that any maximum stable set in a tree contains at least one pendant vertex. Our main finding is the theorem claiming that if T does not own a perfect matching, then at least two pendant vertices an even distance apart belong to core(T). Wh…
Characterizing tricyclic graphs with pendant vertices having largest $A_α$-spectral radius
该研究刻画了含k个悬挂点的n阶三圈图中α∈[1/2,1)时具有最大A_α-谱半径的唯一图，并推导了保证含k个悬挂点的图无三圈结构的充分谱条件。
An edge labeling of a connected graph $G = (V, E)$ is said to be local antimagic if it is a bijection $f:E \to\{1,\ldots ,|E|\}$ such that for any pair of adjacent vertices $x$ and $y$, $f^+(x)\not= f^+(y)$, where the induced vertex label $f^+(x)= \sum f(e)$, with $e$ ranging over all the edges incident to $x$. The local antimagic chromatic number of $G$, denoted by $χ_{la}(G)$, is the minimum number of distinct induced vertex labels over all local antimagic labelings of $G$. Let $χ(G)$ be the chromatic number of $G$. In this paper, sharp upper and lower bounds of $χ_{la}(G)$ for $G$ with pend…
Maximizing the index of signed complete graphs with spanning trees on $k$ pendant vertices
本文研究负边由生成树诱导的不平衡符号完全图，确定在悬挂顶点数给定时使最大特征值最大的极值树与符号图结构。
The row left rank of quaternion unit gain graphs in terms of pendant vertices
本文研究四元数单位增益图的行左秩，利用悬挂顶点给出若干下界，并完整刻画了达到这些下界的极值图。
Connectivity keeping pendant extensions of paths in $k$-connected graphs and triangle-free graphs
本研究证明在$k$-连通图（满足最小度条件）和无三角形图中，路径的悬挂扩展$P_m^+(i)$可作为子图被移除后保持$k$-连通性，从而证实Mader猜想对路径悬挂扩展成立。
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。
Connectivity keeping pendant extensions of paths in $k$-connected triangle-free graphs
该研究针对k-连通无三角形图，证明了满足特定最小度条件时存在保持连通性的路及路的悬挂点扩张子图，证实了相关猜想并优化了最小度界。
On the multiplicity of 1 as a Laplacian eigenvalue of a graph
本文研究图的拉普拉斯特征值1的重数，通过约化图方法证明约化树和约化单圈图的重数上界，并完全刻画达到上界的图类。
Let $G$ be a simple graph. A pendant path of $G$ is a path such that one of its end vertices has degree $1$, the other end has degree $\ge3$, and all the internal vertices have degree $2$. Let $p_k(G)$ be the number of pendant paths of length $k$ of $G$, and $q_k(G)$ be the number of vertices with degree $\ge3$ which are an end vertex of some pendant paths of length $k$. Motivated by the problem of characterizing dendritic trees, N. Saito and E. Woei conjectured that any graph $G$ has some Laplacian eigenvalue with multiplicity at least $p_k(G)-q_k(G)$. We prove a more general result for both …
For some positive integer $k$, if the finite cyclic group $\mathbb{Z}_k$ can act freely on a graph $G$, then we say that $G$ is $k$-symmetric. In 1985, Faria showed that the multiplicity of Laplacian eigenvalue 1 is greater than or equal to the difference between the number of pendant vertices and the number of quasi-pendant vertices. But if a graph has a pendant vertex, then it is at most 1-connected. In this paper, we investigate a class of 2-connected $k$-symmetric graphs with a Laplacian eigenvalue 1. We also identify a class of $k$-symmetric graphs in which all Laplacian eigenvalues are i…
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 $V(T)\cap V(T')=S$. …
Internally-disjoint directed pendant Steiner trees with three terminal vertices in Cartesian product digraphs
本文研究了Cartesian积有向图中三个终端顶点的内部不相交悬垂Steiner树的连通度下界，并提出了一种多项式时间算法来寻找达到该下界的树。
The Kemeny's constant $κ(G)$ of a connected undirected graph $G$ can be interpreted as the expected transit time between two randomly chosen vertices for the Markov chain associated with $G$. In certain cases, inserting a new edge into $G$ has the counter-intuitive effect of increasing the value of $κ(G)$. In the current work we identify a large class of graphs exhibiting this "paradoxical" behavior - namely, those graphs having a pair of twin pendant vertices. We also investigate the occurrence of this phenomenon in random graphs, showing that almost all connected planar graphs are paradoxica…
Pendant paths and integral generalized sun graphs
本文证明含至少三条边的垂悬路径的简单图非积分，解决并反驳了关于积分广义日图的两个猜想，并构造了由Pell方程支配的无限积分图族。
Multiplicity of Laplacian eigenvalue 1 of a graph
研究图的拉普拉斯特征值1的重数上界，通过约化操作转化为准悬挂点度为2的约化图，给出树和一般连通图的上界并刻画极图。
An edge labeling of a connected graph $G = (V, E)$ is said to be local antimagic if it is a bijection $f:E \to\{1,\ldots ,|E|\}$ such that for any pair of adjacent vertices $x$ and $y$, $f^+(x)\not= f^+(y)$, where the induced vertex label $f^+(x)= \sum f(e)$, with $e$ ranging over all the edges incident to $x$. The local antimagic chromatic number of $G$, denoted by $χ_{la}(G)$, is the minimum number of distinct induced vertex labels over all local antimagic labelings of $G$. In this paper, the sharp lower bound of the local antimagic chromatic number of a graph with cut-vertices given by pend…
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 …
On distance spectral radius of power hypertrees with given number of pendant paths of fixed length
研究了具有m条边和k个长度为l的pendant路径的r次幂超树中距离谱半径的最大和最小值问题，确定了唯一最优结构。
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.
Trees with extremal Laplacian eigenvalue multiplicity
本文完整刻画了拉普拉斯特征值重数达到悬挂顶点数减一的树，证明其要么为路径，要么所有悬挂顶点间距离满足特定同余条件，并由此给出特征值1达到该极值重数的充要条件。
A graph is called \emph{claw-free} if it contains no induced subgraph isomorphic to $K_{1,3}$. Matthews and Sumner proved that a 2-connected claw-free graph $G$ is hamiltonian if every vertex of it has degree at least $(|V(G)|-2)/3$. At the workshop C\&C (Novy Smokovec, 1993), Broersma conjectured the degree condition of this result can be restricted only to end-vertices of induced copies of $N$ (the graph obtained from a triangle by adding three disjoint pendant edges). Fujisawa and Yamashita showed that the degree condition of Matthews and Sumner can be restricted only to end-vertices of ind…
In this paper, an upper bound on the nullity of signed graphs in terms of the cyclomatic number and the number of pendant vertices is proved, and the corresponding extremal signed graphs are completely characterized.
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.
Let $H_n$ be the cactus obtained from the star $K_{1,n-1}$ by adding $\lfloor \frac{n-1}{2}\rfloor$ independent edges between pairs of pendant vertices. Let $K_{1,n-1}^+$ be the unicyclic graph obtained from the star $K_{1,n-1}$ by appending one edge. In this paper we give alternative proofs of the following results: Among all cacti with $n$ vertices, $H_n$ is the unique cactus whose spectral radius is maximal, and among all unicyclic graphs with $n$ vertices, $K_{1,n-1}^+$ is the unique unicyclic graph whose spectral radius is maximal. We also prove that among all odd-cycle graphs with $n$ ve…
A result of Bapat and Sivasubramanian gives the inertia of the distance squared matrix of a tree. We develop general tools on how pendant vertices and degree 2 vertices affect the inertia of the distance squared matrix and use these to give an alternative proof of this result. We further use these tools to extend this result to certain families of unicyclic graphs, and we explore how far these results can be extended.
The principal ratio of a connected graph, denoted $γ(G)$, is the ratio of the maximum and minimum entries of its first eigenvector. Cioabă and Gregory conjectured that the graph on $n$ vertices maximizing $γ(G)$ is a kite graph: a complete graph with a pendant path. In this paper we prove their conjecture.
The isotropic matroid $M[IAS(G)]$ of a graph $G$ is a binary matroid, which is equivalent to the isotropic system introduced by Bouchet. In this paper we discuss four notions of connectivity related to isotropic matroids and isotropic systems. We show that the isotropic system connectivity defined by Bouchet is equivalent to vertical connectivity of $M[IAS(G)]$, and if $G$ has at least four vertices, then $M[IAS(G)]$ is vertically 5-connected if and only if $G$ is prime (in the sense of Cunningham's split decomposition). We also show that $M[IAS(G)]$ is $3$-connected if and only if $G$ is conn…
We introduce three new cut tree structures of graphs $G$ in which the vertex set of the tree is a partition of $V(G)$ and contractions of tree vertices satisfy sparsification requirements that preserve various types of cuts. Recently, Kawarabayashi and Thorup \cite{Kawarabayashi2015a} presented the first deterministic near-linear edge-connectivity recognition algorithm. A crucial step in this algorithm uses the existence of vertex subsets of a simple graph $G$ whose contractions leave a graph with $\tilde{O}(n/δ)$ vertices and $\tilde{O}(n)$ edges ($n := |V(G)|$) such that all non-trivial min-…
Graphs whose vertices of degree at least 2 lie in a triangle
本文研究NSF图上的DIM与PED问题复杂性，证明其在若干子类上为NP完全，并给出平面正1-in-3SAT三个连通变体的NP完全性结果。
In this paper we prove that for a general tree $T$, if $A$ is T-TP, all the submatrices of $A$ associated with the deletion of pendant vertices are $P$-matrices, and $\det A>0$, then the smallest eigenvalue has an eigenvector signed according to $T$.
For a given graph $H$, a graph $G$ is $H$-linked if, for every injection $φ: V(H) \to V(G)$, the graph $G$ contains a subdivision of $H$ with $φ(v)$ corresponding to $v$, for each $v\in V(H)$. Let $f(H)$ be the minimum integer $k$ such that every $k$-connected graph is $H$-linked. Among graphs $H$ with at least four vertices, the exact value $f(H)$ is only know when $H$ is a path with four vertices or a cycle with four vertices. A kite is graph obtained from $K_4$ by deleting two adjacent edges, i.e., a triangle together with a pendant edge. Recently, Liu, Rolek and Yu proved that every $8$-co…
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
An ordered graph $H$ on $n$ vertices is a graph whose vertices have been labeled bijectively with $\{1,...,n\}$. The ordered Ramsey number $r_<(H)$ is the minimum $n$ such that every two-coloring of the edges of the complete graph $K_n$ contains a monochromatic copy of $H$ such that the vertices in the copy appear in the same order as in $H$. Although some bounds on the ordered Ramsey numbers of certain infinite families of graphs are known, very little is known about the ordered Ramsey numbers of specific small graphs compared to how much we know about the usual Ramsey numbers for these graph…
The conjecture of Beineke and Harary states that for any two vertices which can be separated by $k$ vertices and $l$ edges for $l\geq 1$ but neither by $k$ vertices and $l-1$ edges nor $k-1$ vertices and $l$ edges there are $k+l$ edge-disjoint paths connecting these two vertices of which $k+1$ are internally disjoint. In this paper we consider this conjecture for $l=2$ and any $k\in \mathbb{N}$. Afterwards, we utilize this result to prove that the conjecture holds for all graphs of treewidth at most $3$ and all $k$ and $l$. We also show that it is NP-complete to decide whether two vertices can…
For a graph $G$, $k(G)$ denotes its connectivity. A graph is super connected if every minimum vertex-cut isolates a vertex. Also $k_{1}$-connectivity of a connected graph is the minimum number of vertices whose deletion gives a disconnected graph without isolated vertices. This paper provides bounds for the super connectivity and $k_{1}$-connectivity of the lexicographic product of two graphs.
The visibility graph of a finite set of points in the plane has the points as vertices and an edge between two vertices if the line segment between them contains no other points. This paper establishes bounds on the edge- and vertex-connectivity of visibility graphs. Unless all its vertices are collinear, a visibility graph has diameter at most 2, and so it follows by a result of Plesník (1975) that its edge-connectivity equals its minimum degree. We strengthen the result of Plesník by showing that for any two vertices v and w in a graph of diameter 2, if deg(v) <= deg(w) then there exist deg(…
Four Dominion Growth Regimes in Trees: Forcing, Fibonacci Enumeration, Periodicity, and Stability
本文研究了树中支配集的增长模式，揭示了不同构造方式对支配集数量的影响，并证明了完全二叉树中支配域的周期性规律及稳定性边界。
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.
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 $q_{min}$…
We determine the minimum sum--connectivity index of bicyclic graphs with $n$ vertices and matching number $m$, where $2\le m\le \lfloor\frac{n}{2}\rfloor$, the minimum and the second minimum, as well as the maximum and the second maximum sum--connectivity indices of bicyclic graphs with $n\ge 5$ vertices. The extremal graphs are characterized.
Prime Vertex Labelings of Several Families of Graphs
本文针对循环悬挂星、循环链、棱柱和广义书等新图族，给出了此前未知的素数顶点标号，即用1到n的整数单射标号顶点，使相邻顶点标号互素。
Let $P_n$ and $C_n$ denote the path and cycle on $n$ vertices respectively. The dumbbell graph, denoted by $D_{p,k,q}$, is the graph obtained from two cycles $C_p$, $C_q$ and a path $P_{k+2}$ by identifying each pendant vertex of $P_{k+2}$ with a vertex of a cycle respectively. The theta graph, denoted by $Θ_{r,s,t}$, is the graph formed by joining two given vertices via three disjoint paths $P_{r}$, $P_{s}$ and $P_{t}$ respectively. In this paper, we prove that all dumbbell graphs as well as theta graphs are determined by their Laplacian spectra.
Let $G$ be a (multi)graph of order $n$ and let $u,v$ be vertices of $G$. The maximum number of internally disjoint $u$-$v$ paths in $G$ is denoted by $κ_G(u,v)$, and the maximum number of edge-disjoint $u$-$v$ paths in $G$ is denoted by $λ_G (u,v)$. The average connectivity of $G$ is defined by $\overlineκ(G)=\sum_{\{u,v\}\subseteq V(G)} κ_G(u,v)/\tbinom{n}{2},$ and the average edge-connectivity of $G$ is defined by $\overlineλ(G)=\sum_{\{u,v\}\subseteq V(G)} λ_G(u,v)/\tbinom{n}{2}$. A graph $G$ is called ideally connected if $κ_G(u,v)=\min\{\mathrm{deg}(u),\mathrm{deg}(v)\}$ for all pairs of …
We investigate a graph parameter called the total vertex irregularity strength ($tvs(G)$), i.e. the minimal $s$ such that there is a labeling $w: E(G)\cup V(G)\rightarrow \{1,2,..,s\}$ of the edges and vertices of $G$ giving distinct weighted degrees $wt_G(v):=w(v)+\sum_{v\in e \in E(G)}w(e)$ for every pair of vertices of $G$. We prove that $tvs(F)=\lceil (n_1+1)/2 \rceil$ for every forest $F$ with no vertices of degree 2 and no isolated vertices, where $n_1$ is the number of pendant vertices in $F$. Stronger results for trees were recently proved by Nurdin et al.
Structural and extremal properties of $l_1$-Fiedler value
本文研究了$l_1$-Fiedler值$b(G)$的结构和极值性质，推导了相关不等式和极值构造，并建立了其与拉普拉斯矩阵和等周数的联系。
The Harary index of a graph is defined as the sum of reciprocals of distances between all pairs of vertices of the graph. In this paper we provide an upper bound of the Harary index in terms of the vertex or edge connectivity of a graph. We characterize the unique graph with maximum Harary index among all graphs with given number of cut vertices or vertex connectivity or edge connectivity. In addition we also characterize the extremal graphs with the second maximum Harary index among the graphs with given vertex connectivity.
Let $n \geq 4$. The helm graph $H_n$ on $2n-1$ vertices is obtained from the wheel graph $W_n$ by adjoining a pendant edge to each vertex of the outer cycle of $W_n$. Suppose $n$ is even. Let $D := [d_{ij}]$ be the distance matrix of $H_n$. In this paper, we first show that $\det(D) = 3(n-1)2^{n-1}.$ Next, we find a matrix $Ł$ and a vector $u$ such that \[D^{-1} = -\frac{1}{2}Ł+\frac{4}{3(n-1)}uu'.\] We also prove an interlacing property between the eigenvalues of $Ł$ and $D$.
The chain theorem of Tutte states that every 3-connected graph can be constructed from a wheel $W_n$ by repeatedly adding edges and splitting vertices. It is not difficult to prove the following strengthening of this theorem: every non-wheel 3-connected graph can be constructed from $W_4$ by repeatedly adding edges and splitting vertices. In this paper we similarly strengthen several chain theorems for various versions of 4-connectivity.
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