Laplacian spectral characterization of some double starlike trees

A tree is called double starlike if it has exactly two vertices of degree greater than two. Let $H(p,n,q)$ denote the double starlike tree obtained by attaching $p$ pendant vertices to one pendant vertex of the path $P_n$ and $q$ pendant vertices to the other pendant vertex of $P_n$. In this paper, we prove that $H(p,n,q)$ is determined by its Laplacian spectrum

A lower bound for the first Zagreb index and its application

For a graph G, the first Zagreb index is defined as the sum of the squares of the vertices degrees. By investigating the connection between the first Zagreb index and the first three coefficients of the Laplacian characteristic polynomial, we give a lower bound for the first Zagreb index, and we determine all corresponding extremal graphs. By doing so, we generalize some known results, and, as an application, we use these results to study the Laplacian spectral determination of graphs with small first Zagreb index

On the spectral characterization of the p-sun and the (p,q)-double sun L. Emilio Allem, Lucas G.M. da Silveira e Vilmar Trevisan

In 1973 Schwenk [7] proved that almost every tree has a cospectral mate. Inspired by Schwenk's result, in this paper we study the spectrum of two families of trees. The p-sun of order is a star with an edge attached to each pendant vertex, which we show to be determined by its spectrum among connected graphs. The -double sun of order is the union of a p-sun and a q-sun by adding an edge between their central vertices. We determine when the -double sun has a cospectral mate and when it is determined by its spectrum among connected graphs. Our method is based on the fact that these trees have few distinct eigenvalues and we are able to take advantage of their nullity to shorten the list of candidates

Complementary spectrum of graphs

Neste trabalho, apresentaremos nosso estudo acerca de grafos coespectrais. Mostraremos construções de famílias de grafos coespectrais já conhecidas na literatura e também construções desenvolvidas durante nossa pesquisa envolvendo grafos thresholds e produto cartesiano. Iremos compartilhar com o leitor o processo histórico que envolve questionamentos acerca de grafos coespecrais. Por fim, apresentaremos nossa maior contribuição: sugerimos usar o espectro complementar de um grafo como alternativa para a representação espectral. O espectro complementar não se trata de associar uma nova matriz a um grafo, mas sim de utilizar a já conhecida matriz de adjacências de uma forma diferente. Nesse viés, realizamos experimentos com famílias de grafos já conhecidas como as árvores, por exemplo. O espectro complementar, juntamente com os conceitos de raio espectral e entrelaçamento de grafos deram o suporte e embasamento para nosso estudo. Por fim, estudamos o conceito de matróide e tentamos vincular com nosso problema de coespectralidade de grafos. Encontramos uma aplicação de um conhecido resultado de Teoria de Matróides na Teoria Espectral de Grafos, mais especificamente, na determinação de grafos.In this work, we present our study around cospectral graphs. We display constructions of cospectral graphs already known in the literature, and also some constructions developed in our own research, which involve threshold graphs and cartesian product. Also, we share with the reader the historic process of raising questions about cospectral graphs. Finally, we then present our greatest contribution: we suggest use the complementary spectrum of a graph as an alternative to spectral representation. The complementary spectrum is not about associating a new matrix to a graph, instead it is about utilizing the already known adjacency matrix in a different way. In this bias, we experiment with families of graphs that are well known, such as the trees, for example. The complementary spectrum, along with the concepts of spectral radius and graph interlacing, gave us the support and foundation to our study. In the end, we study the concept of matroids and try to tie it with our problem of graph cospectrality. We find an application of a known result of the Matroid Theory on the Spectral Graph Theory, specifically, on graph determination

Which Graphs are Determined by their Spectrum?

AMS classifications; 05C50; 05E30;

Symmetries of Nonlinear PDEs on Metric Graphs and Branched Networks

This Special Issue focuses on recent progress in a new area of mathematical physics and applied analysis, namely, on nonlinear partial differential equations on metric graphs and branched networks. Graphs represent a system of edges connected at one or more branching points (vertices). The connection rule determines the graph topology. When the edges can be assigned a length and the wave functions on the edges are defined in metric spaces, the graph is called a metric graph. Evolution equations on metric graphs have attracted much attention as effective tools for the modeling of particle and wave dynamics in branched structures and networks. Since branched structures and networks appear in different areas of contemporary physics with many applications in electronics, biology, material science, and nanotechnology, the development of effective modeling tools is important for the many practical problems arising in these areas. The list of important problems includes searches for standing waves, exploring of their properties (e.g., stability and asymptotic behavior), and scattering dynamics. This Special Issue is a representative sample of the works devoted to the solutions of these and other problems

Some Graph Laplacians and Variational Methods Applied to Partial Differential Equations on Graphs

In this dissertation we will be examining partial differential equations on graphs. We start by presenting some basic graph theory topics and graph Laplacians with some minor original results. We move on to computing original Jost graph Laplacians of friendly labelings of various finite graphs. We then continue on to a host of original variational problems on a finite graph. The first variational problem is an original basic minimization problem. Next, we use the Lagrange multiplier approach to the Kazdan-Warner equation on a finite graph, our original results generalize those of Dr. Grigor’yan, Dr. Yang, and Dr. Lin. Then we do an original saddle point approach to the Ahmad, Lazer, and Paul resonant problem on a finite graph. Finally, we tackle an original Schrödinger operator variational problem on a locally finite graph inspired by some papers written by Dr. Zhang and Dr. Pankov. The main keys to handling this difficult breakthrough Schrödinger problem on a locally finite graph are Dr. Costa’s definition of uniformly locally finite graph and the locally finite graph analog Dr. Zhang and Dr. Pankov’s compact embedding theorem when a coercive potential function is used in the energy functional. It should also be noted that Dr. Zhang and Dr. Pankov’s deeply insightful Palais-Smale and linking arguments are used to inspire the bulk of our original linking proof