Cybele T. M. Vinagre

Laplacian energy of diameter 3 trees

Abstract Let T n be the set of all trees of diameter 3 and n vertices. We show that the Laplacian energy of any tree in T n is strictly between the Laplacian energy of the path P n and the star S n , partially proving the conjecture that this holds for any tree. We also give a total order by the Laplacian energy in T n . Moreover, we show that this order depends only on the algebraic connectivity of the tree: the Laplacian energy increases as the algebraic connectivity decreases in T n .

Constructing pairs of equienergetic and non-cospectral graphs

Abstract The energy of a simple graph G is the sum of the absolute values of the eigenvalues of its adjacency matrix. Two graphs of the same order are said to be equienergetic if they have the same energy. Several ways to construct equienergetic non-cospectral graphs of very large size can be found in the literature. The aim of this work is to construct equienergetic non-cospectral graphs of small size. In this way, we first construct several special families of such graphs, using the product and the cartesian product of complete graphs. Afterwards, we show how one can obtain new pairs of equienergetic non-cospectral graphs from the starting ones. More specifically, we characterize the connected graphs G for which the product and the cartesian product of G and K 2 are equienergetic non-cospectral graphs and we extend Balakrishnan’s result: For a non-trivial graph G , G ⊗ C 4 and G ⊗ K 2 ⊗ K 2 are equienergetic non-cospectral graphs, given in [R. Balakrishnan, The energy of a graph, Linear Algebra Appl. 387 (2004) 287–295].

Spectra and Laplacian spectra of arbitrary powers of lexicographic products of graphs

Consider two graphs G and H. Let Hk[G] be the lexicographic product of Hk and G, where Hk is the lexicographic product of the graph H by itself k times. In this paper, we determine the spectrum of Hk[G] and Hk when G and H are regular and the Laplacian spectrum of Hk[G] and Hk for G and H arbitrary. Particular emphasis is given to the least eigenvalue of the adjacency matrix in the case of lexicographic powers of regular graphs, and to the algebraic connectivity and the largest Laplacian eigenvalues in the case of lexicographic powers of arbitrary graphs. This approach allows the determination of the spectrum (in case of regular graphs) and Laplacian spectrum (for arbitrary graphs) of huge graphs. As an example, the spectrum of the lexicographic power of the Petersen graph with the googol number (that is, 10100 ) of vertices is determined. The paper finishes with the extension of some well known spectral and combinatorial invariant properties of graphs to its lexicographic powers.

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.

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.

A note on (gDF)-spaces

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