The Factorization of Adjoint Polynomials of E^G（i）-class Graphs and Chromatically Equivalence Analysis

Let Sn be the star with n vertices, and let G be any connected graph with p vertices. We denote by Eτp＋（r-1）^G（i） the graph obtained from Sr and rG by coinciding the i-th vertex of G with the vertex of degree r - 1 of S,, while the i-th vertex of each component of （r - 1）G be adjacented to r - 1 vertices of degree 1 of St, respectively. By applying the properties of adjoint polynomials, We prove that factorization theorem of adjoint polynomials of kinds of graphs Eτp＋（r-1）^G（i）∪（r - 1）K1 （1 ≤i≤p）. Furthermore, we obtain structure characteristics of chromatically equivalent graphs of their complements.

Signless Laplacian Characteristic Polynomials of Complete Multipartite Graphs

For a simple graph G,let matrix Q(G)=D(G) + A(G) be it's signless Laplacian matrix and Q G (λ)=det(λI Q) it's signless Laplacian characteristic polynomial,where D(G) denotes the diagonal matrix of vertex degrees of G,A(G) denotes its adjacency matrix of G.If all eigenvalues of Q G (λ) are integral,then the graph G is called Q-integral.In this paper,we obtain that the signless Laplacian characteristic polynomials of the complete multi-partite graphs G=K(n_1,n_2,···,n_t).We prove that the complete t-partite graphs K(n,n,···,n)t are Q-integral and give a necessary and sufficient condition for the complete multipartite graphs K(m,···,m)s(n,···,n)t to be Q-integral.We also obtain that the signless Laplacian characteristic polynomials of the complete multipartite graphs K(m,···,m,)s1(n,···,n,)s2(l,···,l)s3.

On the Hosoya index of graphs

Let G be a （molecular） graph. The Hosoya index Z（G） of G is defined as the number of subsets of the edge set E（G） in which no two edges are adjacent in G, i.e., Z（G） is the total number of matchings of G. In this paper, we determine all the connected graphs G with n ＋ 1 ≤ Z（G） ≤5n - 17 for n ≥ 19. As a byproduct, the graphs of n vertices with Hosoya index from the second smallest value to the twenty first smallest value are obtained for n ≥ 19.

Rainbow Vertex-connection Number of Ladder and Mbius Ladder

A vertex-colored graph G is said to be rainbow vertex-connected if every two vertices of G are connected by a path whose internal vertices have distinct colors, such a path is called a rainbow path. The rainbow vertex-connection number of a connected graph G, denoted by rvc(G), is the smallest number of colors that are needed in order to make G rainbow vertex-connected. If for every pair u, v of distinct vertices, G contains a rainbow u-v geodesic, then G is strong rainbow vertex-connected. The minimum number k for which there exists a k-vertex-coloring of G that results in a strongly rainbow vertex-connected graph is called the strong rainbow vertex-connection number of G, denoted by srvc(G). Observe that rvc(G) ≤ srvc(G) for any nontrivial connected graph G. In this paper, for a Ladder L_n,we determine the exact value of srvc(L_n) for n even. For n odd, upper and lower bounds of srvc(L_n) are obtained. We also give upper and lower bounds of the(strong) rainbow vertex-connection number of Mbius Ladder.

The Relation on the Coefficients and Roots of Adjoint Polynomial and Its Application

The parameter R（G） is the function about the front three coeffcients of the adjoint polynomial of graph G. In the paper, the range of R（G） is given when β（G） 〈 β（Dn）, where β（G） is the minimum root of the adjoint polynomial of graph G and the chromatically equivalent classification of tDn is completely depicted.Furthermore, a sufficient and necessary condition for the class of graphs to be chromatically unique is obtained.

On Chromatically Equivalence of a Class of Graphs

A class of new graphs is defined. A sufficient condition for pairs of these graphs to be chromatically equivalent is proven.

The Annealed Entropy of Wiener Number on Random Double Hexagonal Chains

We study a random planar honeycomb lattice model, namely the random double hexagonal chains. This is a lattice system with nonperiodic boundary condition. The Wiener number is an important molecular descriptor based on the distances, which was introduced by the chemist Harold Wiener in 1947. By applying probabilistic method and combinatorial techniques we obtained an explicit analytical expression for the expected value of Wiener number of a random double hexagonal chain, and the limiting behaviors on the annealed entropy of Wiener number when the random double hexagonal chain becomes infinite in length are analyzed.

On the A-extended Lie Rinehart Algebras

The purpose of this paper is to give a brief introduction to the category of Lie Rinehart algebras and introduces the concept of smooth manifolds associated with a unitary, commutative, associative algebra A. It especially shows that the A-extended algebra as well as the action algebra can be realized as the space of A-left invariant vector fields on a Lie group, analogous to the well known relationship of Lie algebras and Lie groups.

Weakly δ-continuous Correspondences on Super-space

In the paper,we define weakly δ-continuous correspondences on super-space,On the basis of δ-open(closed) sets,θ-open(closed) sets and regular open(closed) sets in topological space,some equivalent conditions of this kind of correspondences are obtained,and some applications of subset nets and convergence nets are given.

Graph Description by the Chromatic Polynomials

By means of the chromatic polynomials, this paper provided a necessary and sufficient condition for the graph G being a mono-cycle graph（the Theorem 1）, a first class hi-cycle graph and a second class bicycle graph（the Theorem 2）, respectively.

On the Trees Whose Hosoya Indices Belong to a Closed Interval

Let n and d be two positive integers.By B_(n,d) we denote the graph obtained by identifying an endvertex of path P_d with the center of star S_(n-d+1),where n ≥ d + 1.By C_(n,d) we denote the graph obtained by identifying an endvertex of P_(d-1) with the center of Stare S_(n-d),and the other endvertex of P_(d-1) with the center of S_3 where n ≥ d + 3.By E_(n,d,k) we denote the graph obtained by identifying the vertex v_k of P(v_1 - v_2 - ··· - v_(d+1)) with the center of S_(n-d).In this paper,we completely characterize all trees T which have diameter at least d(d ≥ 3) and satisfy the following conditions:(i) Z(B_(n,d)) ≤ Z(T) ≤ Z(E_(n,d,3)) for n = d + 3;(ii) Z(B_(n,d)) ≤ Z(T) ≤ Z(C_(n,d)) for n ≥ d + 4.

A NEW METHOD TO FIND CHROMATIC POLYNOMIAL OF GRAPH AND ITS APPLICATIONS

Let G be a simple graph, and let f(G, t) be its chromatic polynomial. We

STRUCTURE AND CHARACTERISTICS OF LEFT CLIFFORD SEMIGROUPS

As generalization of Clifford semigroups, left Clifford semigroups are defined and ξ-products for such semigroups and their semilattice decompositions are studied. In particular, considering how a semilattice decomposition becomes a strong semilattice decomposition and ξ-product becomes spined product, some structure theorems and characteristics for this class of semigroups are obtained.

ORE-TYPE CONDITIONS ON PANCYCLISM OF HAMILTONIAN GRAPHS

In this letter, all graphs will be simple; most of the graph terminologies used here can be found in standard texts except those we don’t define. We know that if G is a graph of order n, and d(x)+d(y)≥n whenever xy(?) E(G), x, y∈V(G), then G is a Hamiltonian (Ore. 1960). Further, G is either

BINDING NUMBER OF A GRAPH AND ITS TRIANGLE

The binding number of a graph G, bind (G), is the largest real number c such

THE BINDING NUMBER OF A GRAPH AND ITS PANCYCLISM

The binding number of a simple graph G is denoted by bind (G), it is the largest number c such

Chromaticity of Complete 6-Partite Graphs with Certain Star or Matching Deleted Ⅱ

Let P（G, λ） be the chromatic polynomial of a graph G. Two graphs G and H are said to be chromatically equivalent, denoted G N H, if P（G, λ） = P（H, λ）. We write [G] = {HIH - G}. If [G] = {G}, then G is said to be chromatically unique. In this paper, we first characterize certain complete 6-partite graphs with 6rid＋1 vertices according to the number of 7-independent partitions of G. Using these results, we investigate the chromaticity of G with certain star or matching deleted. As a by-product, many new families of chromatically unique complete 6-partite graphs with certain star or matching deleted are obtained.

SOME NOTES ON HAMILTONIAN GRAPHS

In this letter, all graphs will be simple. The terminologies and notations, which we do not define, are standard. 1. A graph is called K1,3-free graph if the graph does not contain the induced subgraph isomorphic to K1,3. Recently, on characterizing Hamiltonian graphs by forbidden subgraphs,the K1,3-free graph is studied in many ways. A great many interesting results have been obtained. In this letter, we prove the

GRAPH DECOMPOSITION WITH CONSTRAINT ON THE MINIMUM DEGREE

Ⅰ. BACKGROUND AND NOTATIONS In this paper all graphs will be finite, undirected, and have no loops or multiple edges, namely, simple graphs. E. Gyovi put the following question: Is there a (the smallest) natural number f(s,t) such that the vertex set of each graph of connectivity being f (s, t) at least has a decomposition into sets which would induce subgraphs of connectivity being

CONNECTED AND LOCALLY CONNECTED GRAPHS WITH NO INDUCED CLAWS ARE VERTEX-PANCYCLIC

In this letter, all graphs will be finite, undirected, and have no loops or multiple edges. V(G) and E(G) denote respectively the vertex set and the edge set of the graph G. If S(?)V(G), then we denote by G[S] the subgraph induced by S in G. For the vertex u∈V(G), the neighborhood N(u) of u is the set of all vertices of G adjacent to u.