In this paper, we study some extremal problems of three kinds of spectral radii of $$k$$k-uniform hypergraphs (the adjacency spectral radius, the signless Laplacian spectral radius and the incidence $$Q$$Q-spectral radius). We call a connected and acyclic $$k$$k-uniform hypergraph a supertree. We introduce the operation of "moving edges" for hypergraphs, together with the two special cases of this operation: the edge-releasing operation and the total grafting operation. By studying the perturbation of these kinds of spectral radii of hypergraphs under these operations, we prove that for all these three kinds of spectral radii, the hyperstar $$\mathcal {S}_{n,k}$$Sn,k attains uniquely the maximum spectral radius among all $$k$$k-uniform supertrees on $$n$$n vertices. We also determine the unique $$k$$k-uniform supertree on $$n$$n vertices with the second largest spectral radius (for these three kinds of spectral radii). We also prove that for all these three kinds of spectral radii, the loose path $$\mathcal {P}_{n,k}$$Pn,k attains uniquely the minimum spectral radius among all $$k$$k-th power hypertrees of $$n$$n vertices. Some bounds on the incidence $$Q$$Q-spectral radius are given. The relation between the incidence $$Q$$Q-spectral radius and the spectral radius of the matrix product of the incidence matrix and its transpose is discussed.

The extremal spectral radii of $$k$$k-uniform supertrees

Recent empirical and theoretical works on collective behaviors based on a topological interaction are beginning to offer some explanations as for the physical reasons behind the selection of a particular number of nearest neighbors locally affecting each individual's dynamics. Recently, flocking starlings have been shown to topologically interact with a very specific number of neighbors, between six to eight, while metric-free interactions were found to govern human crowd dynamics. Here, we use network- and graph-theoretic approaches combined with a dynamical model of locally interacting self-propelled particles to study how the consensus reaching process and its dynamics are influenced by the number k of topological neighbors. Specifically, we prove exactly that, in the absence of noise, consensus is always attained with a speed to consensus strictly increasing with k. The analysis of both speed and time to consensus reveals that, irrespective of the swarm size, a value of k ~ 10 speeds up the rate of convergence to consensus to levels close to the one of the optimal all-to-all interaction signaling. Furthermore, this effect is found to be more pronounced in the presence of environmental noise.

/pdf/influence-of-the-number-of-topologically-interacting-2eukjp7amy.pdf

Influence of the number of topologically interacting neighbors on swarm dynamics

https://dspace.stir.ac.uk/bitstream/1893/18445/1/graphs%20for%20which.pdf

Graphs for which the least eigenvalue is minimal, II

Solvability of fractional three-point boundary value problems with nonlinear growth☆

In this paper, we study the p-Laplacian model involving the Caputo fractional derivative with Dirichlet-Neumann boundary conditions. Using a fixed point theorem, we prove the existence of at least three solutions of the model. As an application, an example is included to illustrate the main results.

/pdf/multiple-solutions-of-a-p-laplacian-model-involving-a-3ydwcbmdd8.pdf

Multiple solutions of a p-Laplacian model involving a fractional derivative

The spectral radius of trees on k pendant vertices

A -uniform hypergraph is called odd-bipartite, if is even and there exists some proper subset of such that each edge of contains odd number of vertices in . Odd-bipartite hypergraphs are generalizations of the ordinary bipartite graphs. We study the spectral properties of the connected odd-bipartite hypergraphs. We prove that the Laplacian H-spectrum and signless Laplacian H-spectrum of a connected -uniform hypergraph are equal if and only if is even and is odd-bipartite. We further give several spectral characterizations of the connected odd-bipartite hypergraphs. We also give a characterization for a connected -uniform hypergraph whose Laplacian spectral radius and signless Laplacian spectral radius are equal; thus, provide an answer to a question raised by L. Qi. By showing that the Cartesian product of two odd-bipartite -uniform hypergraphs is still odd-bipartite, we determine that the Laplacian spectral radius of is the sum of the Laplacian spectral radii of and , when and are both connected odd-bipa...

Some spectral properties and characterizations of connected odd-bipartite uniform hypergraphs

Existence and uniqueness of solution for fractional differential equations with integral boundary conditions.

A $k$-uniform hypergraph $G=(V,E)$ is called odd-bipartite ([5]), if $k$ is even and there exists some proper subset $V_1$ of $V$ such that each edge of $G$ contains odd number of vertices in $V_1$. Odd-bipartite hypergraphs are generalizations of the ordinary bipartite graphs. We study the spectral properties of the connected odd-bipartite hypergraphs. We prove that the Laplacian H-spectrum and signless Laplacian H-spectrum of a connected $k$-uniform hypergraph $G$ are equal if and only if $k$ is even and $G$ is odd-bipartite. We further give several spectral characterizations of the connected odd-bipartite hypergraphs. We also give a characterization for a connected $k$-uniform hypergraph whose Laplacian spectral radius and signless Laplacian spectral radius are equal, thus provide an answer to a question raised in [9]. By showing that the Cartesian product $G\Box H$ of two odd-bipartite $k$-uniform hypergraphs is still odd-bipartite, we determine that the Laplacian spectral radius of $G\Box H$ is the sum of the Laplacian spectral radii of $G$ and $H$, when $G$ and $H$ are both connected odd-bipartite.

Some Spectral Properties and Characterizations of Connected Odd-bipartite Uniform Hypergraphs

In this paper, we consider a generalized join operation, that is, the H-join on graphs, where H is an arbitrary graph. In terms of the signless Laplacian and the normalized Laplacian, we determine the spectra of the graphs obtained by this operation on regular graphs. Some additional consequences on the spectral radius, integral graphs and cospectral graphs, etc. are also obtained.

Baofeng Wu

Papers

The spectral radius of trees on k pendant vertices

Some spectral properties and characterizations of connected odd-bipartite uniform hypergraphs

Existence and uniqueness of solution for fractional differential equations with integral boundary conditions.

Some Spectral Properties and Characterizations of Connected Odd-bipartite Uniform Hypergraphs

Signless Laplacian and normalized Laplacian on the H-join operation of graphs