Abstract:
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.

Abstract:
In 1997 Klav\v{z}ar and Gutman suggested a generalization of the Wiener index to vertex-weighted graphs. We minimize the Wiener index over the set of trees with the given vertex weights' and degrees' sequences and show an optimal tree to be the, so-called, Huffman tree built in a bottom-up manner by sequentially connecting vertices of the least weights.

Abstract:
This paper investigates some properties of the number of subtrees of a tree with given degree sequence. These results are used to characterize trees with the given degree sequence that have the largest number of subtrees, which generalizes the recent results of Kirk and Wang. These trees coincide with those which were proven by Wang and independently Zhang et al. to minimize the Wiener index. We also provide a partial ordering of the extremal trees with different degree sequences, some extremal results follow as corollaries.

Abstract:
The Wiener index of a graph is the sum of the distances between all pairs of vertices, it has been one of the main descriptors that correlate achemical compound's molecular graph with experimentally gathered data regarding the compound's characteristics. The tree that minimizes the Wiener index among trees of given maximal degree was studied. We characterize trees that achieve the maximum and minimum Wiener index, given the number of vertices and the degree sequence.

Abstract:
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 $\Delta$.

Abstract:
Let $\lambda_1,\dots,\lambda_n$ be the eigenvalues of a graph $G$. For any $k\geq 0$, the $k$-th spectral moment of $G$ is defined by $\M_k(G)=\lambda_1^k+\dots+\lambda_n^k$. We use the fact that $\M_k(G)$ is also the number of closed walks of length $k$ in $G$ to show that among trees $T$ whose degree sequence is $D$ or majorized by $D$, $\M_k(T)$ is maximized by the greedy tree with degree sequence $D$ (constructed by assigning the highest degree in $D$ to the root, the second-, third-, \dots highest degrees to the neighbors of the root, and so on) for any $k\geq 0$. Several corollaries follow, in particular a conjecture of Ili\'c and Stevanovi\'c on trees with given maximum degree, which in turn implies a conjecture of Gutman, Furtula, Markovi\'c and Gli\v{s}i\'c on the Estrada index of such trees, which is defined as $\EE(G)=e^{\lambda_1}+\dots+e^{\lambda_n}$.

Abstract:
In this paper, we provide algorithms to rank, unrank, and randomly generate certain degree-restricted classes of Cayley trees. Specifically, we consider classes of trees that have a given degree sequence or a given multiset of degrees. If the underlying set of trees have n vertices, then the largest ranks involved in each case are of order n! so that it takes O(nlog(n)) bits just to write down the ranks. Our ranking and unranking algorithms for these degree-restricted classes are as efficient as can be expected since we show that they require O(n^2log(n)) bit operations if the underlying trees have n vertices.

Abstract:
In this paper, we investigate the structures of an extremal tree which has the minimal number of subtrees in the set of all trees with the given degree sequence of a tree. In particular, the extremal trees must be caterpillar and but in general not unique. Moreover, all extremal trees with a given degree sequence $\pi=(d_1, ..., d_5, 1,..., 1)$ have been characterized.

Abstract:
Let $A(G)$ be the adjacency matrix of graph $G$ with eigenvalues $\lambda_1(G), \lambda_2(G),..., \lambda_n(G)$ in non-increasing order. The number $S_k(G):=\sum_{i=1}^{n}\lambda_i^{k}(G)\, (k=0, 1,..., n-1)$ is called the $k$th spectral moment of $G$. Let $S(G) = (S_0(G), S_1(G),..., S_{n-1}(G))$ be the sequence of spectral moments of $G.$ For two graphs $G_1, G_2$, we have $G_1\prec_{s}G_2$ if for some $k \in \{1,2,3,...,n-1\}$, we have $S_i(G_1) = S_i(G_2)\, ,\, i = 0, 1,..., k-1$ and $S_k(G_1)

Abstract:
A proper vertex-colouring of a graph G is said to be locally identifying if for any pair u,v of adjacent vertices with distinct closed neighbourhoods, the sets of colours in the closed neighbourhoods of u and v are different. We show that any graph G has a locally identifying colouring with $2\Delta^2-3\Delta+3$ colours, where $\Delta$ is the maximum degree of G, answering in a positive way a question asked by Esperet et al. We also provide similar results for locally identifying colourings which have the property that the colours in the neighbourhood of each vertex are all different and apply our method to the class of chordal graphs.