A Frequency Assignment Problem (FAP) is the problem that arises when to a given set of transmitters frequencies have to be assigned such that spectrum is used efficiently and the interference between the transmitters is minimal. In this paper we see the frequency assignment problem as a generalised graph colouring problem, where transmitters are presented by vertices and interaction between two...

Detecting protein complexes is an important way to discover the relationship between network topological structure and its functional features in protein-protein interaction (PPI) network. The spectral clustering method is a popular approach. However, how to select its optimal Laplacian matrix is still an open problem. Here, we analyzed the performances of three graph Laplacian matrices (unnorm...

The Laplacian spread of a graph is defined as the difference between the largest and second smallest eigenvalues of the Laplacian matrix of the graph. In this paper, bounds are obtained for the Laplacian spread of graphs. By the Laplacian spread, several upper bounds of the Nordhaus-Gaddum type of Laplacian eigenvalues are improved. Some operations on Laplacian spread are presented. Connected c...

Progress in development of multi-agent control is reviewed. Different approaches for control, estimation, and optimization are discussed a systematic way with particular emphasis on the graph-theoretic perspective. Attention paid to design systems via Laplacian dynamics, as well role graph spectrum, challenges unbalanced digraphs, consensus-based estimation statistics. Some emergent issues, e.g...

Abstract Graphs can be associated with a matrix according to some rule and we find the spectrum of graph respect that matrix. Two graphs are cospectral if they have same spectrum. Constructions help us establish patterns about structural information not preserved by We generalize construction for previously given distance Laplacian larger family graphs. In addition, show appropriate assumptions...

We consider magnetic Schrödinger operators on quantum graphs with identical edges. The spectral problem for the quantum graph is reduced to the discrete magnetic Laplacian on the corresponding combinatorial graph and a certain Hill equation. This may be viewed as a generalization of the classical spectral analysis for the Hill operator to such structures. Using this correspondence we show that ...

We give a method to construct cospectral graphs for the normalized Laplacian by a local modification in some graphs with special structure. Namely, under some simple assumptions, we can replace a small bipartite graph with a cospectral mate without changing the spectrum of the entire graph. We also consider a related result for swapping out biregular bipartite graphs for the matrix A + tD. We p...

For a finite commutative ring $\mathbb{Z}_{n}$ with identity $1\neq 0$, the zero divisor graph $\Gamma(\mathbb{Z}_{n})$ is simple connected having vertex set as of non-zero divisors, where two vertices $x$ and $y$ are adjacent if only $xy=0$. We find distance Laplacian spectrum graphs for different values $n$. Also, we obtain $n=p^z$, $z\geq 2$, in terms spectrum. As consequence, determine thos...

This is an elementary introduction to the Hodge Laplacian on a graph, a higher-order generalization of the graph Laplacian. We will discuss basic properties including cohomology and Hodge theory. At the end we will also discuss the nonlinear Laplacian on a graph, a nonlinear generalization of the graph Laplacian as its name implies. These generalized Laplacians will be constructed out of coboun...