arXiv subjects
Publications and source records attributed to Xiao-Dong Zhang.
It is well known that the complete multipartite graphs can not be determined by their adjacency spectra. But in this paper, we prove that they can be determined by their distance spectra, which confirms the conjecture proposed by Lin, Hong, Wang and Shu.
This paper investigates some relationship between the algebraic connectivity and the clique number of graphs. We characterize all extremal graphs which have the maximum and minimum the algebraic connectivity among all graphs of order $n$ with the clique number $r$, respectively. In turn, an upper and lower bounds for the clique number of a graph in terms of the algebraic connectivity are obtained. Moreover, a spectral version of the Erdős-Stone theorem in terms of the algebraic connectivity of graphs is presented.
Let $G_w$ be a weighted graph. The \textit{inertia} of $G_w$ is the triple $In(G_w)=\big(i_+(G_w),i_-(G_w), $ $ i_0(G_w)\big)$, where $i_+(G_w),i_-(G_w),i_0(G_w)$ are the number of the positive, negative and zero eigenvalues of the adjacency matrix $A(G_w)$ of $G_w$ including their multiplicities, respectively. $i_+(G_w)$, $i_-(G_w)$ is called the \textit{positive, negative index of inertia} of $G_w$, respectively. In this paper we present a lower bound for the positive, negative index of weighted unicyclic graphs of order $n$ with fixed girth and characterize all weighted unicyclic graphs attaining this lower bound. Moreover, we characterize the weighted unicyclic graphs of order $n$ with two positive, two negative and at least $n-6$ zero eigenvalues, respectively.
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 $Δ$.
A new algebraic Cayley graph is constructed using finite fields. Its connectedness and diameter bound are studied via Weil's estimate for character sums. These graphs provide a new source of expander graphs, extending classical results of Chung.
The Wiener index of a graph, which is the sum of the distances between all pairs of vertices, has been well studied. Recently, Sills and Wang in 2012 proposed two conjectures on the maximal Wiener index of trees with a given degree sequence. This note proves one of the two conjectures and disproves the other.
Let $G$ be an unicyclic graph of order $n$ and let $Q_G(x)= det(xI-Q(G))={matrix} \sum_{i=1}^n (-1)^i φ_i x^{n-i}{matrix}$ be the characteristic polynomial of the signless Laplacian matrix of a graph $G$. We give some transformations of $G$ which decrease all signless Laplacian coefficients in the set $\mathcal{G}(n,m)$. $\mathcal{G}(n,m)$ denotes all n-vertex unicyclic graphs with matching number $m$. We characterize the graphs which minimize all the signless Laplacian coefficients in the set $\mathcal{G}(n,m)$ with odd (resp. even) girth. Moreover, we find the extremal graphs which have minimal signless Laplacian coefficients in the set $\mathcal{G}(n)$ of all $n$-vertex unicyclic graphs with odd (resp. even) girth.
Let $G$ be a graph of order $n$ and $Q_G(x)= det(xI-Q(G))= \sum_{i=1}^n (-1)^i φ_i x^{n-i}$ be the characteristic polynomial of the signless Laplacian matrix of a graph $G$. We give some transformations of $G$ which decrease all signless Laplacian coefficients in the set $\mathcal{B}(n)$ of all $n$-vertex bicyclic graphs. $\mathcal{B}^1(n)$ denotes all n-vertex bicyclic graphs with at least one odd cycle. We show that $B_n^1$ (obtained from $C_4$ by adding one edge between two non-adjacent vertices and adding $n-4$ pendent vertices at the vertex of degree 3) minimizes all the signless Laplacian coefficients in the set $\mathcal{B}^1(n)$. Moreover, we prove that $B_n^2$ (obtained from $K_{2,3}$ by adding $n-5$ pendent vertices at one vertex of degree 3) has minimum signless Laplacian coefficients in the set $\mathcal{B}^2(n)$ of all $n$-vertex bicyclic graphs with two even cycles.
When considering the number of subtrees of trees, the extremal structures which maximize this number among binary trees and trees with a given maximum degree lead to some interesting facts that correlate to other graphical indices in applications. The number of subtrees in the extremal cases constitute sequences which are of interest to number theorists. The structures which maximize or minimize the number of subtrees among general trees, binary trees and trees with a given maximum degree have been identified previously. Most recently, results of this nature are generalized to trees with a given degree sequence. In this note, we characterize the trees which maximize the number of subtrees among trees of a given order and degree sequence. Instead of using theoretical arguments, we take an algorithmic approach that explicitly describes the process of achieving an extremal tree from any random tree. The result also leads to some interesting questions and provides insight on finding the trees close to extremal and their numbers of subtrees.
Gold nanoparticles have been conceived as a radiosensitizer in cancer radiation therapy, but one of the important questions for primary drug screening is what size of gold nanoparticles can optimally enhance radiation effects. Herein, we perform in vitro and in vivo radiosensitization studies of 4.8, 12.1, 27.3, and 46.6 nm PEG-coated gold nanoparticles. In vitro results show that all sizes of the PEG-coated gold nanoparticles can cause a significant decrease in cancer cell survival after gamma radiation. 12.1 and 27.3 nm PEG-coated gold nanoparticles have dispersive distributions in the cells and have stronger sensitization effects than 4.8 and 46.6 nm particles by both cell apoptosis and necrosis. Further, in vivo results also show all sizes of the PEG-coated gold nanoparticles can decrease tumor volume and weight after 5 Gy radiations, and 12.1 and 27.3 nm PEG-coated gold nanoparticles have greater sensitization effects than 4.8 and 46.6 nm particles, which can lead to almost complete disappearance of the tumor. In vivo biodistribution confirms that 12.1 and 27.3 nm PEG-coated gold nanoparticles are accumulated in the tumor with high concentrations. The pathology, immune response, and blood biochemistry indicate that the PEG-coated gold nanoparticles do not cause spleen and kidney damages, but give rise to liver damage and gold accumulation. It can be concluded that 12.1 and 27.3 nm PEG-coated gold nanoparticles show high radiosensitivity, and these results have an important indication for possible radiotherapy and drug delivery.
Gold nanoparticles have shown great prospective in cancer diagnosis and therapy, but they can not be metabolized and prefer to accumulate in liver and spleen due to their large size. The gold nanoclusters with small size can penetrate kidney tissue and have promise to decrease in vivo toxicity by renal clearance. In this work, we explore the in vivo renal clearance, biodistribution, and toxicity responses of the BSA- and GSH-protected gold nanoclusters for 24 hours and 28 days. The BSA-protected gold nanoclusters have low-efficient renal clearance and only 1% of gold can be cleared, but the GSH-protected gold nanoclusters have high-efficient renal clearance and 36 % of gold can be cleared after 24 hours. The biodistribution further reveals that 94% of gold can be metabolized for the GSH-protected nanoclusters, but only less than 5% of gold can be metabolized for the BSA-protected nanoclusters after 28 days. Both of the GSH- and BSA-protected gold nanoclusters cause acute infection, inflammation, and kidney function damage after 24 hours, but these toxicity responses for the GSH-protected gold nanoclusters can be eliminated after 28 days. Immune system can also be affected by the two kinds of gold nanoclusters, but the immune response for the GSH-protected gold nanoclusters can also be recovered after 28 days. These findings show that the GSH-protected gold nanoclusters have small size and can be metabolized by renal clearance and thus the toxicity can be significantly decreased. The BSA- protected gold nanoclusters, however, can form large compounds and further accumulate in liver and spleen which can cause irreparable toxicity response. Therefore, the GSH-protected gold nanoclusters have great potential for in vivo imaging and therapy, and the BSA-protected gold nanoclusters can be used as the agent of liver cancer therapy.
In this paper, we present a sharp upper and lower bounds for the signless Laplacian spectral radius of graphs in terms of clique number. Moreover, the extremal graphs which attain the upper and lower bounds are characterized. In addition, these results disprove the two conjectures on the signless Laplacian spectral radius in [P. Hansen and C. Lucas, Bounds and conjectures for the signless Laplacian index of graphs, Linear Algebra Appl., 432(2010) 3319-3336].
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 $π=(d_1, ..., d_5, 1,..., 1)$ have been characterized.
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.
We present the GGA+U calculations to investigate the electronic structure and visible light absorption of the N, B-codoped anatase TiO2. The NsBi (substitutional N, interstitial B) codoped TiO2 produces significant Ti 3d and N 2p mid-gap states when the distance of N and B atoms is far, and the NiBi (interstitial N and B) and NsBs (substitutional N and B) codoped TiO2 prefer to form localized p states at 0.3-1.2 eV above the valence band maximum. Further, the optical band edges of the three codoped systems shift slightly to the visible region, but only the far distance NsBi codoped TiO2 shows an obvious visible optical transition. These results indicate that the NsBi codoped TiO2 has a dominant contribution to the visible absorption of the N, B-codoped TiO2.
Since some realistic networks are influenced not only by increment behavior but also by tunable clustering mechanism with new nodes to be added to networks, it is interesting to characterize the model for those actual networks. In this paper, a weighted local-world model, which incorporates increment behavior and tunable clustering mechanism, is proposed and its properties are investigated, such as degree distribution and clustering coefficient. Numerical simulations are fit to the model characters and also display good right skewed scale-free properties. Furthermore, the correlation of vertices in our model is studied which shows the assortative property. Epidemic spreading process by weighted transmission rate on the model shows that the tunable clustering behavior has a great impact on the epidemic dynamic. Keywords: Weighted network, increment behavior, tun- able cluster, epidemic spreading.
It is difficult to detect and evaluate the number of communities in complex networks, especially when the situation involves with an ambiguous boundary between the inner- and inter-community densities. In this paper, Discrete Nodal Domain Theory could be used to provide a criterion to determine how many communities a network would have and how to partition these communities by means of the topological structure and geometric characterization. By capturing the signs of certain Laplacian eigenvectors we can separate the network into several reasonable clusters. The method leads to a fast and effective algorithm with application to a variety of real networks data sets.
It is well-known that the pancake graphs are widely used as models for interconnection networks \cite{Akers}. In this paper, some properties of the pancake graphs are investigated. We first prove that the pancake graph, denoted by $P_n (n\geq 4),$ is super-connected and hyper-connected. Further, we study the symmetry of $P_n$ and completely determine its full automorphism group,which shows that $P_n (n\geq 5)$ is a graphical regular representation of $S_n.$