Invasive sequence and ecological niche of main insect borers of Larix gmelinii forest in Aershan, Inner Mongolia

After the outbreak of Dendrolimus superans Buter in 2002, many insect borers quickly invaded larch （Larix gmelinii Rupr.） forests in the Aershan of Inner Mongolia. Methods involved included setting sample plots, collecting adults in iron traps and measuring areas of galleries to study the invasive sequence, their ecological niche and the extent of the different effects by the main insect borers to their hosts. The results showed that the damage of D. superans weakened L. gmelinii, first Ips subelongatus Motschulsky invaded, followed by Acanthocinus carinulatus Gebler, Monochamus urussovi Fisher and M. sutor L. After the outbreak of D. superans, the average density of longhorn beetles per L. gmelinii tree increased. The ecological niche of Ips subelongatus stretches almost from the base to the top of the trunk. The number of insects in older stands of L. gmelinii is larger than those in middle aged stands. They do not damage healthy trees of L. gmelinii. The ecological niche of A. carinulatus is higher in dead L. gmelinii trees than in weak ones. The degree of damage is directly proportional with age and depth of bark. M. urussovi mainly damages trunks below 4 m in weak trees; in dead trees they can do damage up to 6 m in height. M. sutor mainly damages trunks below 5 m in weak L. gmelinii trees; in dead trees they cause damage up to 7 m. Again, the degree of damage is directly proportional with age. None of the three species of longhorn beetles damage healthy L. gmelinii and younger trees. Among the main insect borers, the degree of damage caused by I. subelongatus is more serious than that of other insects.

Characterizing C6＋P2-graphic Sequences

For a given graph H, a graphic sequence π =(d1, d2, ···, dn) is said to be potentially H-graphic if π has a realization containing H as a subgraph. In this paper, we characterize the potentially C6+ P2-graphic sequences where C6+ P2 denotes the graph obtained from C6 by adding two adjacent edges to the three pairwise nonadjacent vertices of C6. Moreover, we use the characterization to determine the value of σ(C6+ P2, n).

The Smallest Degree Sum That Yields Potentially Kr＋1 - K3-Graphic Sequences

Let a（Kr,＋1 - K3,n） be the smallest even integer such that each n-term graphic sequence п= （d1,d2,…dn） with term sum σ（п） = d1 ＋ d2 ＋…＋ dn 〉 σ（Kr＋1 -K3,n） has a realization containing Kr＋1 - K3 as a subgraph, where Kr＋1 -K3 is a graph obtained from a complete graph Kr＋1 by deleting three edges which form a triangle. In this paper, we determine the value σ（Kr＋1 - K3,n） for r ≥ 3 and n ≥ 3r＋ 5.

Spectral radius of uniform hypergraphs and degree sequences

We present several upper bounds for the adjacency and signless Laplacian spectral radii of uniform hypergraphs in terms of degree sequences.

Multigraphic Degree Sequences and Hamiltonian-connected Line Graphs

Let G be a multigraph.Suppose that e=u1v1 and e′=u2v2 are two edges of G.If e≠e′,then G(e,e′)is the graph obtained from G by replacing e=u1v1 with a path u1vev1 and by replacing e′=u2v2 with a path u2ve′v2,where ve,ve′are two new vertices not in V(G).If e=e′,then G(e,e′),also denoted by G(e),is obtained from G by replacing e=u1v1 with a path u1vev1.A graph G is strongly spanning trailable if for any e,e′∈E(G),G(e,e′)has a spanning(ve,ve′)-trail.The design of n processor network with given number of connections from each processor and with a desirable strength of the network can be modelled as a degree sequence realization problem with certain desirable graphical properties.A sequence d=(d1,d2,⋯,dn)is multigraphic if there is a multigraph G with degree sequence d,and such a graph G is called a realization of d.A multigraphic degree sequence d is strongly spanning trailable if d has a realization G which is a strongly spanning trailable graph,and d is line-hamiltonian-connected if d has a realization G such that the line graph of G is hamiltonian-connected.In this paper,we prove that a nonincreasing multigraphic sequence d=(d1,d2)⋯,dn)is strongly spanning trailable if and only if either n=1 and d1=0 or n≥2 and dn≥3.Applying this result,we prove that for a nonincreasing multigraphic sequence d=(d1,d2,⋯,dn),if n≥2 and dn≥3,then d is line-hamiltonian-connected.

An Evolving Random Network and Its Asymptotic Structure

In this paper, we propose an evolving random network. The model is a linear combination of preferential attachment model and uniform model. We show that scaling limit distribution of the number of leaves at time n is approximated by nomal distribution and the proportional degree sequence obeys power law. The branching structure and maximum degree are also discussed in this paper.

Exact Solution to an Extremal Problem on Graphic Sequences with a Realization Containing Every 2-Tree on k Vertices

A simple graph G is a 2-tree if G=K3,or G has a vertex v of degree 2,whose neighbors are adjacent,and G-v is a 2-tree.Clearly,if G is a 2-tree on n vertices,then |E(G)|=2 n-3.A non-increasing sequence π=(d1,...,dn) of nonnegative integers is a graphic sequence if it is realizable by a simple graph G on n vertices.[Acta Math.Sin.Engl.Ser.,25,795-802(2009)] proved that if k≥2,n≥9/2 k^22+19/2 k and π=(d1,...,dn) is a graphic sequence with∑i=1^n di>(k-2)n,then π has a realization containing every 1-tree(the usual tree) on k vertices.Moreover,the lower bound(k-2)n is the best possible.This is a variation of a conjecture due to Erdos and Sos.In this paper,we investigate an analogue problem for 2-trees and prove that if k≥3 is an integer with k≡i(mod 3),n≥ 20[k/3] 2+31[k/3]+12 and π=(d1,...,dn) is a graphic sequence with ∑i=1^n di>max{k-1)(n-1), 2 [2 k/3] n-2 n-[2 k/3] 2+[2 k/3]+1-(-1)i}, then π has a realization containing every 2-tree on k vertices.Moreover,the lower bound max{(k-1)(n-1), 2[2 k/3]n-2 n-[2 k/3] 2+[2 k/3]+1-(-1)i}is the best possible.This result implies a conjecture due to [Discrete Math.Theor.Comput.Sci.,17(3),315-326(2016)].

A Characterization for a Sequence to be Potentially K_(r+1) — e-graphic

Let n 〉 r, let lr --- （dl,d2,-,dn） be a non-increasing sequence of nonnegative integers and let Kr＋l - e be the graph obtained from Kr＋l by deleting one edge. If zr has a realization G containing Kr＋l - e as a subgraph, then r is said to be potentially Kr＋l - e-graphic. In this paper, we give a characterization for a sequence π to be potentially Kr＋l - e-graphic.

The Degree Distribution of the Random Multigraphs

In this paper, as a generalization of the binomial random graph model, we define the model of multigraphs as follows： let G（n; {pk}） be the probability space of all the labelled loopless multigraphs with vertex set V = {v1, v2, ..., vn }, in which the distribution of tvi,vj, the number of the edges between any two vertices vi and vj is P{tvi,vj =k}=Pk, k=0, 1,2,...and they are independent of each other. Denote by Xd = Xd（G）,Yd = Yd（G）, Zd = Zd（G） and Zcd = Zcd（G） the number of vertices of G with degree d, at least d, at most d and between c and d. In this paper, we discuss the distribution of Xd, Yd, Zd and Zcd in the probability space G（n; （Pk））.

A Rao-type Characterization for a Sequence to Have a Realization Containing an Arbitrary Subgraph H

Let G be an arbitrary spanning subgraph of the complete graph Kr＋1 on r＋1 vertices and Kr＋1-E（G） be the graph obtained from Kr＋1 by deleting all edges of G.A non-increasing sequence π=（d1,d2,...,dn） of nonnegative integers is said to be potentially Kr＋1-E（G）-graphic if there is a graph on n vertices that has π as its degree sequence and contains Kr＋1-E（G） as a subgraph.In this paper,a characterization of π that is potentially Kr＋1-E（G）-graphic is given,which is analogous to the Erdo s–Gallai characterization of graphic sequences using a system of inequalities.This is a solution to an open problem due to Lai and Hu.As a corollary,a characterization of π that is potentially Ks,tgraphic can also be obtained,where Ks,t is the complete bipartite graph with partite sets of size s and t.This is a solution to an open problem due to Li and Yin.

Graphic Sequences with a Realization Containing Intersecting Cliques

Let r ≥1, k ≥ 2 and Fm1,… k;r denote the most general definition of a friendship graph,

The Threshold for the Erdos,Jacobson and Lehel Conjecture to Be True

Let σ（k, n） be the smallest even integer such that each n-term positive graphic sequence with term sum at least σ（k, n） can be realized by a graph containing a clique of k ＋ 1 vertices. Erdos et al. （Graph Theory, 1991, 439-449） conjectured that σ（k, n） = （k - 1）（2n- k） ＋ 2. Li et al. （Science in China, 1998, 510-520） proved that the conjecture is true for k 〉 5 and n ≥ （k2） ＋ 3, and raised the problem of determining the smallest integer N（k） such that the conjecture holds for n ≥ N（k）. They also determined the values of N（k） for 2 ≤ k ≤ 7, and proved that [5k-1/2] ≤ N（k） ≤ （k2） ＋ 3 for k ≥ 8. In this paper, we determine the exact values of σ（k, n） for n ≥ 2k＋3 and k ≥ 6. Therefore, the problem of determining σ（k, n） is completely solved. In addition, we prove as a corollary that N（k） -= [5k-1/2] for k ≥6.

A Variation of a Conjecture Due to Erds and Sós

Erdoes and Soes conjectured in 1963 that every graph G on n vertices with edge number e（G） 〉 1/2（k - 1）n contains every tree T with k edges as a subgraph. In this paper, we consider a variation of the above conjecture, that is, for n 〉 9/ 2k^2 ＋ 37/2＋ 14 and every graph G on n vertices with e（G） 〉 1/2 （k- 1）n, we prove that there exists a graph G＇ on n vertices having the same degree sequence as G and containing every tree T with k edges as a subgraph.