The hamiltonian number problem is a generalization of hamiltonian cycle problem in graph theory. It is well known that the hamiltonian cycle problem in graph theory is NP-complete [16]. So the hamiltonian number problem is also NP-complete. On the other hand, the hamiltonian number problem is the traveling salesman problem with each edge having weight 1. A hamiltonian walk of a graph G is a closed spanning walk of minimum length. The length of a hamiltonian walk in G is called the hamiltonian number. For graphs, we give some bounds for hamiltonian numbers of graphs. First, we improve some results in [14] and give a necessary and sufficient condition for h(G) < e(G) where e(G) is the minimum length of a closed walk passing through all edges of G. Next, we prove that if two nonadjacent vertices u and v satisfying that deg(u)+deg(v) ≥ |G|, then h(G) = h(G + uv). This result generalizes a theorem of Bondy and Chv¡¬atal for the hamiltonian cycle. Finally, we show that if 0 ≤ k ≤ n − 2 and G is a 2-connected graph of order n satisfying deg(u) + deg(v) + deg(w) ≥ 3n−k−2 for every independent set {u, v,w} of three vertices in G, then h(G) ≤ n+k. It is a generalization of a Bondy¡¦s result. For digraphs, we give some bounds for hamiltonian numbers of digraphs first. We prove that if a digraph D of order n is strongly connected, thenn ≤ h(D) ≤ ⌊(n+1)^2/4 ⌋. Next, we also present some digraphs of order n ≥ 5 which have hamiltonian number k for n ≤ k ≤ ⌊(n+1)^2/4 ⌋. Finally, we study hamiltonian numbers of M¡Lobius double loop networks. We introduce M¡Lobius double loop network and every strongly connected double loop network is isomorphic to some M¡Lobius double loop network. Next, we give an upper bound for the hamiltonian numbers of M¡Lobius double loop networks. Then, we find some necessary and sufficient conditions for M¡Lobius double loop networks MDL(d, m, ℓ) to have hamiltonian numbers dm, dm + 1 or dm + 2.

給定一個圖形G，以及集合M，M為一描述圖形G中各點擁有訊息之情形的集合。圖形G相對於M的的傳遞數是指，於最短時間內，讓圖形中全部點皆獲得所有種類之訊息，並將符號記為t(G;M) 。傳遞過程中每個時間單位將受到下列限制: (1)圖形上的每個點只能與自己相鄰的點交換訊息。 (2)兩個相鄰的點在每個單位時間裡至多只能交換一個訊息。 我們希望可以找到在最短的時間裡完成傳遞的方法，也就是讓圖形G中的每一個點都獲得所有種類之訊息，我們稱此類型問題為訊息傳遞問題。 在本論文中，給定一個圖形G，且圖形G中每個點的訊息只有一個，G中任兩點的訊息都不會相同，符號t(G)代表完成傳遞所需最少的時間單位。我們給定圖形的傳遞數的上界與下界，並且定出一套公式計算樹圖、完全二部圖及雙環網路圖的傳遞數。 / Given a graph G together with a set M , the transmission number of G corresponding to M , denoted by t(G;M), is the minimum number of time needed to complete the transmission , that is, to let all the vertices in G know all the messages in M , subject to the constraints that at each time unit, each vertex can interchange messages with all its neighbors, but the number of messages that two vertices can interchange at each time unit is at most one. We want to find the minimum number of time units required to complete the transmission, that is, to let all the vertices in G know all the messages. We call such a problem the message transmission problem. Given a graph G, the transmission number of G, denoted t(G), is the minimum number of time units required to complete the transmission, under the condition that |m(v)|=1 for all v in V(G). In this thesis, we give upper and lower bounds for the transmission number of G, and give formulas to compute the transmission numbers of trees, complete bipartite graphs and double loop networks.