Narrow Results

Let $G$ be a connected graph and $A(G)$ be the adjacency matrix of $G$. Suppose that ${\lambda }_1(G)\ge {\lambda }_2(G)\ge \cdots \ge {\lambda }_n(G)$ are the eigenvalues of $G$. In this paper, we first give a graft transformation on the spectral radius of graphs and then as their application, we determine the extremal graphs with maximum and minimum spectral radii among all clique trees. Furthermore, we also determine the unique graph with maximum spectral radius among all block graphs by using different methods.

An outerplanar graph is a graph which can be embedded in the plane so that all vertices are on one face, say the outer face. Let $C_k$ be a cycle of length $k$ with $k\ge 3$. Among the set of outerplanar graphs on $n$ vertices without $C_k$, we first obtain that the extremal graph having the maximum spectral radius contains a subgraph $S_n$ for $k\ge 3$ and $n\ge 551227$ and then we deduce the unique extremal graph having the maximum spectral radius for $n>max\{{(2+6\times 2^{\frac{k}{2}+1})}^2,551227\}$ and $k\ge 4$, where $S_n$ is the star with $n$ vertices. Furthermore, for $n\ge 9$, among the set of outerplanar graphs without $C_3$ on $n$ vertices, we get that the star $S_n$ is the unique extremal graph having the maximum spectral radius.

For a graph $G=(V(G),E(G))$, the sum of the distance between vertex $u\in V(G)$ and other vertices in $G$ is denoted by ${\sigma }_G(u)$. The Balaban index of $G$ is defined as $J(G)=\frac{m}{\mu +1}{\sum }_{uv\in E(G)}\frac{1}{\sqrt{{\sigma }_G(u){\sigma }_G(v)}}$, where $\mu =m-n+1$, $m=|E(G)|$, $n=|V(G)|$. Based on the Balaban index of a graph, a weighted function is defined as $f(x,y)=\frac{1}{\sqrt{xy}}$, $x>0,y>0$. Define a matrix $A_f(G)$ as the weighted adjacency matrix of $G$, where $A_f(G)(u,v)=f({\sigma }_G(u),{\sigma }_G(v))$ if $u\sim v$ and $A_f(G)(u,v)=0$ otherwise. In this work, we show that star $S_n$ is with the largest spectral radius of $A_f(T)$ among trees with $n$ vertices. Inversely, path $P_n$ is with the smallest spectral radius of $A_f(T)$ in all caterpillar trees with $n$ vertices.

We apply spectral radius and norm inequalities to certain $2\times 2$ operator matrices to give simple proofs, refinements and generalizations of known norm inequalities. New norm inequalities are also given. Our analysis uncovers the interplay between different spectral radius and norm inequalities.

In this paper, we investigate analytic symbols $\psi$ and $\phi$ when the weighted composition operator $C_{\psi ,\phi }$ is complex symmetric on general function space $H_s$. As applications, we characterize completely the compactness, normality and isometry of complex symmetric weighted composition operators. Especially, we show that the equivalence of compactness and Hilbert–Schmidtness, and the existence of non-normal complex symmetric operators for such operators, which answers one open problem raised by Noor in [On an example of a complex symmetric composition operators on $H^2(𝔻)$, J. Funct. Anal.269 (2015) 1899–1901] for higher dimensional case.

The inappropriate network structure often facilitates the epidemic spreading driven by traffic. In this paper, an edge reconnection strategy based on node closeness centrality is proposed to decrease the speed of epidemic spreading. In simulations, some evaluation indicators are introduced to measure the optimized effect. After adopting the proposed strategy, the spectral radius of optimized network gradually decreases as the number of edge reconnection increases, and eventually reaches a stable value. Consequently, the epidemic threshold, as a significant indicator to evaluate the speed of epidemic spreading, shows a noticeable increase. In addition, the comparisons of average path length and the scale of epidemic spreading can be intuitively obtained.

We consider a problem in Mathematical Biology that leads to a question in Graph Theory, which can be solved using an old but not widely known upper estimate of the spectral radius of a non-negative matrix. We provide a new proof of this estimate.

Let L≀X be a lamplighter graph, i.e., the graph-analogue of a wreath product of groups, and let P be the transition operator (matrix) of a random walk on that structure. We explain how methods developed by Saloff-Coste and the author can be applied for determining the ℓ^p-norms and spectral radii of P, if one has an amenable (not necessarily discrete or unimodular) locally compact group of isometries that acts transitively on L. This applies, in particular, to wreath products K≀G of finitely-generated groups, where K is amenable. As a special case, this comprises a result of Żuk regarding the ℓ²-spectral radius of symmetric random walks on such groups.

In this paper, we compute the Hausdorff dimension of a graph-directed set when the underlying multigraph is a Cartesian product or a tensor product of several multigraphs. We give explicit formulas in terms of the eigenvalues of the graph and the similarity ratios used with each graph.

The HHT-α method previously developed shows favorable numerical dissipation, in addition to unconditional stability, in the solution of linear elastic systems. However, its performance in the solution of nonlinear systems has not been studied yet. In this paper, the numerical properties of a subfamily of this integration method with -⅓ ≤ α ≤ 0, β = ¼(1 - α)² and γ = ½-α for the solution of nonlinear systems are analytically explored using a linearized analysis, where the stiffness term in total form is used to determine the restoring force. A theoretical proof of unconditional stability for nonlinear systems is presented. The subfamily of the integration method is verified to possess favorable numerical dissipation for both linear and nonlinear systems. Period distortion affected by the step degree of either nonlinearity or convergence is studied. Although the analysis is conducted for a single-degree-of-freedom nonlinear system, the application to a multiple-degree-of-freedom nonlinear system is also illustrated. It is confirmed that the performance of the HHT-α method for nonlinear systems is generally the same as that for linear elastic systems, except for the high dependence on the step degree of nonlinearity.

In [M. M. Malakiyeh, S. Shojaee and S. Hamzehei-Javaran, Development of a direct time integration method based on Bezier curve and 5th-order Berstein basis function, Comput. Struct. 194 (2108) 15–31] an unconditionally stable implicit time-integration method using the Bezier curve was proposed for solving structural dynamic problems. In this study, a new class of the previous algorithm is presented by using the Bernstein polynomials and the Bezier curve as the interpolation functions for solving the equations of motion with the possibility of using large time steps. The spectral radius, period elongation, amplitude decay and overshooting of the present method are investigated and compared with some other methods. To show the high-performance, robustness and validity of this method, five numerical examples are presented. The theoretical analysis and numerical examples show that the proposed method has low dissipation in the lower modes and high dissipation in the higher modes in comparison with the other methods reported in the literature.

The distance signless Laplacian matrix $D^Q(G)$ of a connected graph $G$ is defined as $D^Q(G)=\mathrm{\text{Tr}}(G)+D(G)$, where $D(G)$ is the distance matrix of $G$ and $\mathrm{\text{Tr}}(G)$ is the diagonal matrix whose main entries are the vertex transmissions of $G$, and the spectral radius of a connected graph $G$ is the largest eigenvalue of $D^Q(G)$. In this paper, first we obtain the $D^Q(G)$-eigenvalues of the join of certain regular graphs. Next, we give some new bounds on the distance signless Laplacian spectral radius of a graph $G$ in terms of graph parameters and characterize the extremal graphs. Utilizing these results we present some upper and lower bounds on the distance signless Laplacian energy of a graph $G$.

For a simple connected graph $G$, the reciprocal transmission $T{r}_G^{\prime }(v)$ of a vertex $v$ is defined as

Let $G$ be a graph with $n$ vertices and let $d_i$ be the degree of its $i$th vertex ($d_i$ is the degree of $v_i$), then the Zagreb matrix of $G$ is the square matrix of order $n$ whose $(i,j)$-entry is equal to $d_i+d_j$ if the $i$th and $j$th vertex of $G$ are adjacent, and zero otherwise. We give some bounds for the Zagreb spectral radius in terms of the maximum degree and minimum degree of $G$, the Randić index $R_{-1/2}$, and the first Zagreb index $M_1$. We also obtain some bounds for the Zagreb Estrada index.

In this paper, we determine the two graphs in the class of unicyclic bipartite graphs with n vertices, having the first two largest spectral radius.

The signless Laplacian matrix of a graph is the sum of its degree diagonal and adjacency matrices. In this paper, we present a sharp upper bound for the spectral radius of the adjacency matrix of a graph. Then this result and other known results are used to obtain two new sharp upper bounds for the signless Laplacian spectral radius. Moreover, the extremal graphs which attain an upper bound are characterized.

Let be the set of unicyclic graphs with n vertices and k cut vertices. In this paper, we determine the unique graph with the maximal spectral radius among all graphs in for 1 ≤ k ≤ n - 3.

The independence number of a graph is defined as the maximum size of a set of pairwise non-adjacent vertices and the spectral radius is defined as the maximum eigenvalue of the adjacency matrix of the graph. Xu et al. in [The minimum spectral radius of graphs with a given independence number, Linear Algebra and its Applications431 (2009) 937–945] determined the connected graphs of order n with independence number which minimize the spectral radius. In this paper, we show that the graph obtained from a path of order α by blowing up each vertex to a clique of order k minimizes the spectral radius among all connected graphs of order kα with independence number α for α = 3, 4 and conjecture that this is true for all α ∈ ℕ.

In a simple connected graph, the average 2-degree of a vertex is the average degree of its neighbors. With the average 2-degree sequence and the maximum degree ratio of adjacent vertices, we present a sharp upper bound of the spectral radius of the adjacency matrix of a graph, which improves a result in [Y. H. Chen, R. Y. Pan and X. D. Zhang, Two sharp upper bounds for the signless Laplacian spectral radius of graphs, Discrete Math. Algorithms Appl.3(2) (2011) 185–191].

Let $\mathcal{𝒞}(C_n,W_n)$ denote the set of all weighted cycles with vertex set $V(C_n)=\{v_0,v_1,\dots ,v_{n-1}\}$, edge set $E(C_n)=\{v_i{v}_j|j-i\equiv \pm 1\mathrm{\text{mod}}n\}$ and positive weight set $W_n=\{w_1\ge w_2\ge \cdots \ge w_n>0\}$. A weighted cycle $G^{\ast }\in \mathcal{𝒞}(C_n,W_n)$ is called maximum if ${\lambda }_1(G^{\ast })\ge {\lambda }_1(G)$ for any $G\in \mathcal{𝒞}(C_n,W_n)$. In this paper, we give some properties of the Perron vector for the maximum weighted graphs and then determine the maximum weighted cycle in $\mathcal{𝒞}(C_n,W_n)$.