Renata R. Del-Vecchio

A note on a conjecture for the distance Laplacian matrix

In this note, the graphs of order n having the largest distance Laplacian eigenvalue of multiplicity n− 2 are characterized. In particular, it is shown that if the largest eigenvalue of the distance Laplacian matrix of a connected graph G of order n has multiplicity n − 2, then G = Sn or G = Kp,p, where n = 2p. This resolves a conjecture proposed by M. Aouchiche and P. Hansen in [M. Aouchiche and P. Hansen. A Laplacian for the distance matrix of a graph. Czechoslovak Mathematical Journal, 64(3):751–761, 2014.]. Moreover, it is proved that if G has P5 as an induced subgraph then the multiplicity of the largest eigenvalue of the distance Laplacian matrix of G is less than n− 3.

Diagonalization of generalized lollipop graphs

Abstract Let G be a graph formed by connecting a tree and a threshold graph with an edge between their respective roots. Let A be the adjacency matrix of G and x ∈ R . We give an O ( n ) algorithm for constructing a diagonal matrix D congruent to A + x I n , allowing us to locate eigenvalues of A and obtain spectral results.

On Q-spectral integral variation

Abstract Let G be a connected graph with two nonadjacent vertices and G ′ be the graph constructed from G by adding an edge between them. It is known that the trace of Q ′ is 2 plus the trace of Q, where Q and Q ′ are the signless Laplacian matrices of G and G ′ , respectively. Hence, the sum of the eigenvalues of Q ′ is the sum of the eigenvalues of Q plus 2. Since none of the eigenvalues of Q can decrease if an edge is added to G, it is said that Q-spectral integral variation occurs when either only one Q-eigenvalue is increased by 2, or when two Q-eigenvalues are increased by 1 one each. In this article we give necessary and sufficient conditions for the occurrence of Q-spectral integral variation only in two places, as the first case never occurs.

Cograph generation with linear delay

Abstract Cographs have always been a research target in areas such as coloring, graph decomposition, and spectral theory. In this work, we present an algorithm to generate all unlabeled cographs with n vertices, based on the generation of cotrees. The delay of our algorithm (time spent between two consecutive outputs) is O ( n ) . The time needed to generate the first output is also O ( n ) , which gives an overall O ( n M n ) time complexity, where M n is the number of unlabeled cographs with n vertices. The algorithm avoids the generation of duplicates (isomorphic outputs) and produces, as a by-product, a linear ordering of unlabeled cographs with n vertices.

Laplacian integrality in P4-sparse and P4-extendible graphs

Let G be a simple graph and L=L(G) the Laplacian matrix of G. G is called L-integral if all its Laplacian eigenvalues are integer numbers. It is known that every cograph, a graph free of P4, is L-integral. The class of P4-sparse graphs and the class of P4-extendible graphs contain the cographs. It seems natural to investigate if the graphs in these classes are still L-integral. In this paper we characterized the L-integral graphs for both cases, P4-sparse graphs and P4-extendible graphs.

Hyper-Hamiltonicity in graphs: some sufficient conditions

Abstract A Hamiltonian graph G is hyper-Hamiltonian if G − v is Hamiltonian for any v ∈ V ( G ) . In this paper, we give some sufficient conditions for a graph to be hyper-Hamiltonian. We provide both, spectral and non-spectral conditions for hyper-Hamiltonicity.

Construction of a Molecular Model: A Mathematical-Chemical Interdisciplinary Approach in the Secondary Education

This paper presents the construction of molecular models from easily accessible materials as well as details of the geometrical and trigonometric theories that support it. Its aim is to allow a chemical-mathematical interdisciplinary approach to the subject, under the assumption that knowledge of the molecular geometrical shape, as well as of biand three-dimensional representations in chemistry, is supplied by the linguistic complements provided by the overlap between the theories of chemistry and mathematics.

A note on (gDF)-spaces

Certain locally convex spaces of scalar-valued mappings are shown to be finite- dimensional.