For $0leq alpha < 1$, the $mathcal{A}_{alpha}$-spectral radius of a $k$-uniform hypergraph $G$ is defined to be the spectral radius of the tensor $mathcal{A}_{alpha}(G):=alpha mathcal{D}(G)+(1-alpha) mathcal{A}(G)$, where $mathcal{D}(G)$ and $A(G)$ a
re diagonal and the adjacency tensors of $G$ respectively. This paper presents several lower bounds for the difference between the $mathcal{A}_{alpha}$-spectral radius and an average degree $frac{km}{n}$ for a connected $k$-uniform hypergraph with $n$ vertices and $m$ edges, which may be considered as the measures of irregularity of $G$. Moreover, two lower bounds on the $mathcal{A}_{alpha}$-spectral radius are obtained in terms of the maximum and minimum degrees of a hypergraph.
Let $F_{a_1,dots,a_k}$ be a graph consisting of $k$ cycles of odd length $2a_1+1,dots, 2a_k+1$, respectively which intersect in exactly a common vertex, where $kgeq1$ and $a_1ge a_2ge cdotsge a_kge 1$. In this paper, we present a sharp upper bound fo
r the signless Laplacian spectral radius of all $F_{a_1,dots,a_k}$-free graphs and characterize all extremal graphs which attain the bound. The stability methods and structure of graphs associated with the eigenvalue are adapted for the proof.
In this paper, we present two sharp upper bounds for the spectral radius of (bipartite) graphs with forbidden a star forest and characterize all extremal graphs. Moreover, the minimum least eigenvalue of the adjacency matrix of graph with forbidden a
star forest and all extremal graphs for graphs are obtained.
The distance energy of a simple connected graph $G$ is defined as the sum of absolute values of its distance eigenvalues. In this paper, we mainly give a positive answer to a conjecture of distance energy of clique trees proposed by Lin, Liu and Lu [
H.~Q.~ Lin, R.~F.~Liu, X.~W.~Lu, The inertia and energy of the distance matrix of a connected graph, {it Linear Algebra Appl.,} 467 (2015), 29-39.]
This paper is concerned with the extreme points of the polytopes of stochastic tensors. By a tensor we mean a multi-dimensional array over the real number field. A line-stochastic tensor is a nonnegative tensor in which the sum of all entries on each
line (i.e., one free index) is equal to 1; a plane-stochastic tensor is a nonnegative tensor in which the sum of all entries on each plane (i.e., two free indices) is equal to 1. In enumerating extreme points of the polytopes of line- and plane-stochastic tensors of order 3 and dimension $n$, we consider the approach by linear optimization and present new lower and upper bounds. We also study the coefficient matrices that define the polytopes.
The Steiner distance of vertices in a set $S$ is the minimum size of a connected subgraph that contain these vertices. The sum of the Steiner distances over all sets $S$ of cardinality $k$ is called the Steiner $k$-Wiener index and studied as the nat
ural generalization of the famous Wiener index in chemical graph theory. In this paper we study the extremal structures, among trees with a given segment sequence, that maximize or minimize the Steiner $k$-Wiener index. The same extremal problems are also considered for trees with a given number of segments.
Let $G_1$ and $G_2$ be two simple connected graphs. The invariant textit{coronal} of graph is used in order to determine the $alpha$-eigenvalues of four different types of graph equations that are $G_1 circ G_2, G_1lozenge G_1$ and the other two`s ar
e $G_1 odot G_2$ and $G_1 circleddash G_2$ which are obtained using the $R$-graph of $G_1$. As an application we construct infinitely many pairs of non-isomorphic $alpha$-Isospectral graph.
Tur{a}n type extremal problem is how to maximize the number of edges over all graphs which do not contain fixed forbidden subgraphs. Similarly, spectral Tur{a}n type extremal problem is how to maximize (signless Laplacian) spectral radius over all gr
aphs which do not contain fixed subgraphs. In this paper, we first present a stability result for $kcdot P_3$ in terms of the number of edges and then determine all extremal graphs maximizing the signless Laplacian spectral radius over all graphs which do not contain a fixed linear forest with at most two odd paths or $kcdot P_3$ as a subgraph, respectively.
In this paper, we obtain a sharp upper bound for the spectral radius of a nonnegative matrix. This result is used to present upper bounds for the adjacency spectral radius, the Laplacian spectral radius, the signless Laplacian spectral radius, the di
stance spectral radius, the distance Laplacian spectral radius, the distance signless Laplacian spectral radius of a graph or a digraph. These results are new or generalize some known results.
Radiotherapy is often the most straightforward first line cancer treatment for solid tumors. While it is highly effective against tumors, there is also collateral damage to healthy proximal tissues especially with high doses. The use of radiosensitiz
ers is an effective way to boost the killing efficacy of radiotherapy against the tumor while drastically limiting the received dose and reducing the possible damage to normal tissues. Here, we report the design and application of a good radiosensitizer by using ultrasmall gold nanoclusters with a naturally occurring peptide (e.g., glutathione or GSH) as the protecting shell. The GSH coated gold nanoclusters can escape the RES absorption, leading to a good tumor uptake (8.1% ID/g at 24 h post injection). As a result, the as-designed Au nanoclusters led to a strong enhancement for radiotherapy, as well as a negligible damage to normal tissues. After the treatment, the ultrasmall gold nanoclusters can be efficiently cleared by the kidney, thereby avoiding potential long term side effects caused by the accumulation of gold atoms in the body. Our data suggest that the ultrasmall peptide protected Au nanoclusters are a promising radiosensitizer for cancer radiotherapy.
