The edge-Wiener index of a graph

If G is a connected graph, then the distance between two edges is, by definition, the distance between the corresponding vertices of the line graph of G. The edge-Wiener index W_e of G is then equal to the sum of distances between all pairs of edges of G. We give bounds on W_e in terms of order and size. In particular we prove the asymptotically sharp upper bound for graphs of order n.     

一类三圈图的Wiener指数

图G的Wiener指数是指图G中所有顶点对间的距离之和,即W（G）=∑dc（u,u）,{u,u}CG其中de（u,u）表示G中顶点u,u之间的距离.三圈图是指边数与顶点数之差等于2的连通图,任意两个圈至多只有一个公共点的三圈图记为T_n~3.研究了三圈图T_n~3的Wiener指数,给出了其具有最小、次小Wiener指数的图结构.     

The minimum degree distance of graphs of given order and size

In this note, we study the degree distance of a graph which is a degree analogue of the Wiener index. Given n and e, we determine the minimum degree distance of a connected graph of order n and size e.     

Inverses of -distance matrices of a tree

The determinant and the inverse of the distance matrix of a tree have been investigated in the literature, following the classical formulas due to Graham and Pollak for the determinant, and due to Graham and Lovász for the inverse. We consider two q-analogs of the distance matrix of a tree and obtain formulas for the inverses of the two distance matrices. Yan and Yeh have previously obtained expressions for the determinants of the two distance matrices. Some related results are proved.     

Four new sums of graphs and their Wiener indices

The Wiener index is the sum of distances between all vertex pairs in a connected graph. This notion was motivated by various mathematical properties and chemical applications. In this paper we introduce four new operations on graphs and study the Wiener indices of the resulting graphs.     

On the first passage time of a simple random walk on a tree

We consider a simple random walk on a tree. Exact expressions are obtained for the expectation and the variance of the first passage time, thereby recovering the known result that these are integers. A relationship of the mean first passage matrix with the distance matrix is established and used to derive a formula for the inverse of the mean first passage matrix.     

The small hexagon and heptagon with maximum sum of distances between vertices

The hexagon and heptagon with unit diameter and maximum sum of Euclidean distances between vertices are determined by enumerating diameter configurations, and by using a branch and cut algorithm for nonconvex quadratic programming. Lower bounds on the value on this sum are presented for polygon with a larger number of vertices.     

On the sum of distances along a circle

The arc distance between two points on a circle is their geodesic distance along the circle. We study the sum of the arc distances determined by n points on a circle, which is a useful measure of the evenness of scales and rhythms in music theory. We characterize the configurations with the maximum sum of arc distances by a balanced condition: for each line that goes through the circle center and touches no point, the numbers of points on each side of the line differ by at most one. When the points are restricted to lattice positions on a circle, we show that Toussaint's snap heuristic finds an optimal configuration. We derive closed-form formulas for the maximum sum of arc distances when the points are either allowed to move continuously on the circle or restricted to lattice positions. We also present a linear-time algorithm for computing the sum of arc distances when the points are presorted by the polar coordinates.     

On the roots of Wiener polynomials of graphs

The Wiener polynomial of a connected graph $G$ is defined as $W(G;x)=\sum x^{d(u,v)}$, where $d(u,v)$ denotes the distance between $u$ and $v$, and the sum is taken over all unordered pairs of distinct vertices of $G$. We examine the nature and location of the roots of Wiener polynomials of graphs, and in particular trees. We show that while the maximum modulus among all roots of Wiener polynomials of graphs of order $n$ is $\left(\genfrac{}{}{0ex}{}{n}{2}\right)?1$, the maximum modulus among all roots of Wiener polynomials of trees of order $n$ grows linearly in $n$. We prove that the closure of the collection of real roots of Wiener polynomials of all graphs is precisely $(?\infty ,0]$, while in the case of trees, it contains $(?\infty ,?1]$. Finally, we demonstrate that the imaginary parts and (positive) real parts of roots of Wiener polynomials can be arbitrarily large.     

On the differences between Szeged and Wiener indices of graphs

Let G be a connected graph and η(G)=Sz(G)−W(G), where W(G) and Sz(G) are the Wiener and Szeged indices of G, respectively. A well-known result of Klav?ar, Rajapakse, and Gutman states that η(G)≥0, and by a result of Dobrynin and Gutman η(G)=0 if and only if each block of G is complete. In this paper, a path-edge matrix for the graph G is presented by which it is possible to classify the graphs in which η(G)=2. It is also proved that there is no graph G with the property that η(G)=1 or η(G)=3. Finally, it is proved that, for a given positive integer k,k≠1,3, there exists a graph G with η(G)=k.     

The relation between the number of leaves of a tree and its diameter

Czechoslovak Mathematical Journal - Let L(n, d) denote the minimum possible number of leaves in a tree of order n and diameter d. Lesniak (1975) gave the lower bound B(n,d) = ⌈2(n −...     

Asymptotics for the sum of powers of distances between power residues modulo a prime

For fixed q ∈ (0, 4), prime p → ∞, and \(d \leqslant \exp \left( {c\sqrt {\ln p} } \right)\), where c > 0 is a constant, we obtain the asymptotics for the sum of qth powers of distances between neighboring residues of degree d modulo p.     

Maximizing the distance between center,centroid and characteristic set of a tree

Consider a tree P_n-g,g , n≥ 2, 1≤ g≤ n-1 on n vertices which is obtained from a path on [1,2,?…?,n-g] vertices by adding g pendant vertices to the pendant vertex n-g. We prove that over all trees on n?≥?5 vertices, the distance between center and characteristic set, centroid and characteristic set, and center and centroid is maximized by trees of the form P_n-g,g , 2?≤?g?≤?n-3. For n≥ 5, we also supply the precise location of the characteristic set of the tree P_n-g,g , 2?≤?g?≤?n-3.