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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 observations are discussed and compared with those related to the eccentricity. Then, analogous to the sum of eccentricities we consider the sum of normalities. After briefly discussing the extremal problems in general graphs we focus on trees and trees under various constraints. As opposed to the path and star in numerous extremal problems, we present several interesting and unexpected extremal structures. Lastly we consider the difference between eccentricity and normality, and the sum of these differences. We also introduce some unsolved problems in the context., Comment: 14 pages 9figures

Motivated from the study of eccentricity, center, and sum of eccentricities in graphs and trees, we introduce several new distance-based global and local functions based on the smallest distance from a vertex to some leaf (called the `uniformity' at that vertex). Some natural extremal problems on trees are considered. Then the middle parts of a tree is discussed and compared with the well-known center of a tree. The values of the global functions are also compared with the sum of eccentricities and some sharp bounds are established. Last but not the least, we show that the difference between the eccentricity and the uniformity, when considered as a local function, behaves in a very similar way as the eccentricity itself., Comment: 12 pages

Let $C_{v}(k;T)$ be the number of the closed walks of length $k$ starting at vertex $v$ in a tree $T$. We prove that for a given tree degree sequence $\pi$, then for any tree with degree sequence $\pi$, the sequence $C(k;T)\equiv(C_{v}(k;T), v\in V(T))$ is weakly majorized by the sequence $C(k, T_{\pi}^*)\equiv C(k, T_{\pi}^*, v\in V(T^*))$, where $T_{\pi}^*$ is the greedy tree corresponding to $\pi$. In addition, for two trees degree sequences $\pi,~\pi'$, if $\pi$ is majorized by $\pi'$, then $C(k;T_{\pi}^*)$ is weakly majorized by $C(k;T_{\pi'}^*)$., Comment: 16 pages 2 figures, Applied Mathematics and Computation, 2018

The terminal Wiener index of a tree is the sum of distances for all pairs of pendent vertices, which recently arises in the study of phylogenetic tree reconstruction and the neighborhood of trees. This paper presents a sharp upper and lower bounds for the terminal Wiener index in terms of its order and diameter and characterizes all extremal trees which attain these bounds. In addition, we investigate the properties of extremal trees which attain the maximum terminal Wiener index among all trees of order $n$ with fixed maximum degree., Comment: 15 pages, 1 figure

Heydari \cite{heydari2013} presented very nice formulae for the Wiener and terminal Wiener indices of generalized Bethe trees. It is pity that there are some errors for the formulae. In this paper, we correct these errors and characterize all trees with the minimum terminal Wiener index among all the trees of order $n$ and with maximum degree $\Delta$., Comment: 13 pages

The paper is a brief survey of some recent new results and progress of the Laplacian spectra of graphs and complex networks (in particular, random graph and the small world network). The main contents contain the spectral radius of the graph Laplacian for given a degree sequence, the Laplacian coefficients, the algebraic connectivity and the graph doubly stochastic matrix, and the spectra of random graphs and the small world networks. In addition, some questions are proposed., Comment: 17 pages

In this paper, we investigate how the Wiener index of unicyclic graphs varies with graph operations. These results are used to present a sharp lower bound for the Wiener index of unicyclic graphs of order $n$ with girth and the matching number $\beta\ge \frac{3g}{2}$. Moreover, we characterize all extremal graphs which attain the lower bound., Comment: 15 pages