The eccentricity of a vertex, $ecc_T(v) = \max_{u\in T} d_T(v,u)$, was one of the first, distance-based, tree invariants studied. The total eccentricity of a tree, $Ecc(T)$, is the sum of eccentricities of its vertices. We determine extremal values and characterize extremal tree structures for the ratios $Ecc(T)/ecc_T(u)$, $Ecc(T)/ecc_T(v)$, $ecc_T(u)/ecc_T(v)$, and $ecc_T(u)/ecc_T(w)$ where $u,w$ are leaves of $T$ and $v$ is in the center of $T$.
In addition, we determine the tree structures that minimize and maximize total eccentricity among trees with a given degree sequence.
The Steiner $k$-eccentricity of a vertex $v$ of a graph $G$ is the maximum Steiner distance over all $k$-subsets of $V(G)$ which contain $v$. In this paper Steiner $3$-eccentricity is studied on trees. Some general properties of the Steiner $3$-eccentricity of trees are given. A tree transformation which does not increase the average Steiner $3$-eccentricity is given. As its application, several lower and upper bounds for the average Steiner $3$-eccentricity of trees are derived.
We study the Steiner $k$-eccentricity on trees, which generalizes the previous one in the paper [X.~Li, G.~Yu, S.~Klavžar, On the average Steiner 3-eccentricity of trees, arXiv:2005.10319, 2020]. To support the algorithm, we achieve much stronger properties for the Steiner $k$-ecc tree than that in the previous paper. Based on this, a linear time algorithm is devised to calculate the Steiner $k$-eccentricity of a vertex in a tree. On the other hand, the lower and upper bounds of the average Steiner $k$-eccentricity index of a tree on order $n$ are established based on a novel technique which i…
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…
The eccentricity matrix of a connected graph $G$, denoted by $\mathcal{E}(G)$, is obtained from the distance matrix of $G$ by keeping the largest nonzero entries in each row and each column and leaving zeros in the remaining ones. The eigenvalues of $\mathcal{E}(G)$ are the $\mathcal{E}$-eigenvalues of $G$. The eccentricity energy (or the $\mathcal{E}$-energy) of $G$ is the sum of the absolute values of all $\mathcal{E}$-eigenvalues of $G$. In this article, we determine the unique tree with the minimum second largest $\mathcal{E}$-eigenvalue among all trees on $n$ vertices other than the star.…
The \textit{eccentricity matrix} $\mathcal{E}(G)$ of a connected graph $G$ is obtained from the distance matrix of $G$ by keeping the largest non-zero entries in each row and each column, and leaving zeros in the remaining ones. The eigenvalues of $\mathcal{E}(G)$ are the \textit{$\mathcal{E}$-eigenvalues} of $G$. In this article, we find the inertia of the eccentricity matrices of trees. Interestingly, any tree on more than $4$ vertices with odd diameter has two positive and two negative $\mathcal{E}$-eigenvalues (irrespective of the structure of the tree). A tree with even diameter has the s…
The eccentricity matrix $\varepsilon(G)$ of a graph $G$ is constructed from the distance matrix of $G$ by keeping only the largest distances for each row and each column. This matrix can be interpreted as the opposite of the adjacency matrix obtained from the distance matrix by keeping only the distances equal to 1 for each row and each column. The $\varepsilon$-eigenvalues of a graph $G$ are those of its eccentricity matrix $\varepsilon(G)$. Wang et al \cite{e} proposed the problem of determining the maximum $\varepsilon$-spectral radius of trees with given order. In this paper, we consider t…
The eccentricity matrix $\mathcal E(G)$ of a connected graph $G$ is obtained from the distance matrix of $G$ by leaving unchanged the largest nonzero entries in each row and each column, and replacing the remaining ones with zeros. In this paper, we consider the set $\mathcal C \mathcal T$ of clique trees whose blocks have at most two cut-vertices \textcolor{blue}{of the clique tree}. After proving the irreducibility of the eccentricity matrix of a clique tree in $\mathcal C \mathcal T$ and finding its inertia indices, we show that every graph in $\mathcal C \mathcal T$ with more than $4$ vert…
The eccentricity matrix of a connected graph $G$, denoted by $\mathcal{E}(G)$, is obtained from the distance matrix of $G$ by keeping the largest nonzero entries in each row and each column, and leaving zeros in the remaining ones. The $\mathcal{E}$-eigenvalues of $G$ are the eigenvalues of $\mathcal{E}(G)$, in which the largest one is the $\mathcal{E}$-spectral radius of $G$. The $\mathcal{E}$-energy of $G$ is the sum of the absolute values of all $\mathcal{E}$-eigenvalues of $G$. In this article, we study some of the extremal problems for eccentricity matrices of complements of trees and cha…
The relation between the Wiener index $W(G)$ and the eccentricity $\varepsilon(G)$ of a graph $G$ is studied. Lower and upper bounds on $W(G)$ in terms of $\varepsilon(G)$ are proved and extremal graphs characterized. A Nordhaus-Gaddum type result on $W(G)$ involving $\varepsilon(G)$ is given. A sharp upper bound on the Wiener index of a tree in terms of its eccentricity is proved. It is shown that in the class of trees of the same order, the difference $W(T) - \varepsilon(T)$ is minimized on caterpillars. An exact formula for $W(T) - \varepsilon(T)$ in terms of the radius of a tree $T$ is obt…
本文研究整数的所有合成的和。我们导出了这个量的生成函数，并将其应用于几个特殊函数，包括各种广义伯努利数。我们将合成和与S. G. Woon提出并由P.Fuchs以“通用PI树”的名称进行扩展，其中通过对从$\{g_n\}$构建的树的每一行求和，将输出序列$\{x_n\}$与输入序列$\{g_n\}$相关联。我们的链接与组合物的概念允许引入修改富克斯的树，考虑到非线性变换的生成函数的输入序列。我们还介绍了\textit{generalized sums over compositions}的概念，在那里我们看看组成的每一部分组成的总和。
The Steiner $k$-eccentricity of a vertex $v$ of a graph $G$ is the maximum Steiner distance over all $k$-subsets of $V (G)$ which contain $v$. A linear time algorithm for calculating the Steiner $k$-eccentricity of a vertex on block graphs is presented. For general graphs, an $O(n^{ν(G)+1}(n(G) + m(G) + k))$ algorithm is designed, where $ν(G)$ is the cyclomatic number of $G$. A linear algorithm for computing the Steiner $3$-eccentricities of all vertices of a tree is also presented which improves the quadratic algorithm from [Discrete Appl.\ Math.\ 304 (2021) 181--195].
A broadcast on a nontrivial connected graph G with vertex set V is a function f from V to {0,1,...,diam(G)} such that f(v) is at most the eccentricity of v for all vertices v. The weight of f is the sum of the function values taken over V. A vertex u hears f from v if f(v) is positive and d(u,v) is at most f(v). A broadcast f is boundary independent if, for any vertex w that hears f from vertices v_{1},...,v_{k}, where k is at least 2, d(w,v_{i}) equals f(v_{i}) for each i. The maximum weight of a boundary independent broadcast on G is denoted by α_{bn}(G). We prove a sharp upper bound on α_{b…
A broadcast on a nontrivial connected graph G is a function f from V(G) to the set {0,1,...,diam(G)} such that f(v) is at most the eccentricity of v for all vertices v of G. The weight of f is the sum of the function values over V(G). A vertex u hears f from v if f(v) is positive and u is within distance f(v) from v. A broadcast f is boundary independent if any vertex that hears f from two or more vertices is at distance f(v) from each such vertex v. The maximum weight of a boundary independent broadcast on G is denoted by α_{bn}(G). We prove a sharp lower bound on α_{bn}(T) for a tree T. Comb…
A broadcast on a connected graph $G=(V,E)$ is a function $f:V\rightarrow \{0,1,\dots,\operatorname{diam}(G)\}$ such that $f(v)\leq e(v)$ (the eccentricity of $v$) for all $v\in V$ if $|V|\geq2$, and $f(v)=1$ if $V=\{v\}$. The cost of $f$ is $σ(f)=\sum_{v\in V}f(v)$. Let $V_{f}% ^{+}$ denote the set of vertices $v$ such that $f(v)$ is positive. A vertex $u$ hears $f$ from $v\in V_{f}^{+}$ if the distance $d(u,v)\leq f(v)$. When $f$ is a broadcast such that every vertex $x$ that hears $f$ from more than one vertex in $V_{f}^{+}$ also satisfies $d(x,u)\geq f(u)$ for all $u\in V_{f}^{+}$, we say t…
