This paper has formulated a kind of nonconnected graph C 4∪K m,n ,and proved that the graph C 4∪K m,n is a k graceful graph when k>1(k∈N) ,and that the graph is a ( k,d )arithmetic graph when k>[(n-1)m+1]d+1(d>1 and m,n,d∈N) .

The present paper presents a kind of unconnected graph C 4∪ St (m) , proves that the graph C 4∪ St (m) is a k graceful graph, when k>1, and that the graph is a (k,d) arithmetic graph when k>d+1 and d>1(d∈N).

给出一类非连通图 C4∪ St（m ）． 论证当 k＞ 1（k∈ N ）时， 该图是 k 优美图； 当 k＞ d＋ 1（d ＞ 1， d ∈ N）时， 图 C4∪ St（m ）是（k，d ）算术图．

K-bridge graph is a graph composed of K internally-disjotint paths connecting two vertices A and B.Primitive index of primitive K-bridge graph is calculated.

An integer distance graph is a graph G(Z,D) with the integer set Z as vertex set, in which an edge joining two vertices u and v if and only if u-v∈D , where D is a set of natural numbers.

The calibration graph is A=0.0021C+0.0005, with the linear regression correlative coefficient is 0.9986 and the relative standard deviation (RSD) of 1.21%.

The coefficient a_1(G) of one degree term of chromatic polynomial is studied. A sufficient condition that a connected graph is a planar graph is given. The structure of the graph Gis demonstrated if |a_1(G)|=6,7,8,9,10.

With the reaction being kept running for 8 minutes at room temperature,the determined range of copper is 0.005～0.18mg·L -1 and the regression equation of calibration graph is A=0.2793 c-0.006 with a correlation coefficient of 0.9988.The method is used to determine trace copper in type-water,wild sweet tea and momordica grosvenori,with satisfactory results.

We prove that ′(G)=n for a (n-2) regular graph of ordern,and ′(G)=n-1 for a (n-3) regular graph of order n6, for which complementary graph is a Hamilton cycle.

An integer distance graph is a graph G(D) with the set of all integers Z as vertex set and two adjacent vertices u, v∈Z if and only if |u-v|∈D, where the distance set D is a subset of positive integers.

The conjecture of zero domination of 0-cyclic monotone graphs is proved (anr-cyclic graph is a cyclic monotone (s, t)-graph exactlyr minimal paths of which have cycles).

A covering path in a directed graph is a path passing through all vertices and arcs of the graph, with each arc being traversed only in the direction of its orientation.

A covering path in a directed graph is a path passing through all vertices and arcs of the graph, with each arc being traversed only in the direction of its orientation.

A functional graph is a digraph describing the action of a function on a set.

A generalized Halin graph is a 2-connected plane graph G such that removing all the edges of the boundary of the exterior face of G (the degrees of the vertices in the boundary of exterior face of G are all three) gives a tree.

It is a combinatorial problem with great significance in practics to obtain the bandwidth of a graph. However, it has been shown that algorithmic determination of the bandwidth of a graph is an NP-complete problem even for a tree. So it is attractive to get thc bandwidths of some special, classes, of graphs.In 1976, Dewdney announced in a survery, that the bandwidth of the discrete tori Cm × Cn is an unsolved problem. In this paper, we has solved this problem, i.e., we obtain the following...

It is a combinatorial problem with great significance in practics to obtain the bandwidth of a graph. However, it has been shown that algorithmic determination of the bandwidth of a graph is an NP-complete problem even for a tree. So it is attractive to get thc bandwidths of some special, classes, of graphs.In 1976, Dewdney announced in a survery, that the bandwidth of the discrete tori Cm × Cn is an unsolved problem. In this paper, we has solved this problem, i.e., we obtain the following results:Theorem 1. If m≠n and min(m, n)≥3, then the bandwidth of the discrete tori Cm×Cn is 2 min(m, n);Theorem 2. If n≥3, then the bandwidth of the discrete tori Cn×Cn is 2n-1,

A complementary partitive graph is a bipartite graph G(V', V"; E) with disjoint vertex sets V', V", and an edge set E such that all edges are directed except only one edge, say j=[x, y], is undireeted, and the outgoing degrees are d+(x)=0, d+(y)=0 and d+(v)=1 for all v≠x, y. The following assertions can be easily proved: If G(v', V"; E) is a complementary partitive graph with undireeted edge j, then a pair of eomple mentary partitions P'(E) and P"(E) with respect to j can be constructed by...

A complementary partitive graph is a bipartite graph G(V', V"; E) with disjoint vertex sets V', V", and an edge set E such that all edges are directed except only one edge, say j=[x, y], is undireeted, and the outgoing degrees are d+(x)=0, d+(y)=0 and d+(v)=1 for all v≠x, y. The following assertions can be easily proved: If G(v', V"; E) is a complementary partitive graph with undireeted edge j, then a pair of eomple mentary partitions P'(E) and P"(E) with respect to j can be constructed by the edges incident with each vertex of V' and eaeh vertex of V". Conversely, if Hk has a com-plementary partition with respect to j, then a complementary partitive graph can be constructed.By using the complementary partitive graph defined above. We can ease the proofs of theorems of complementary partitions established by W. K. Chen (1969, 1976) and give a simple criterion to determine whether or not a complementary partition is essen-tial as follows:Theorem A complementary partition is essential if and only if the corresponding com-plementary partitive graph is a connected graph.

Bond graph is a new and powerful tool of system simulation.It has provided a simple and convenient method for establishing a mathematic model of a dynamic system. A digital imitation is made by using a computer program and sending directly the bond graph model into computer without establishing a mathematic model by manual in advance.