20 documents found, page 1 of 2
On spectral invariants of the α-mixed adjacency matrix
Andrade, Enide; Lenes, Eber; Pizarro Pamela; Robbiano, María; Rodríguez, JonnathanLet Gˆ be a mixed graph and α∈[0,1]. Let Dˆ(Gˆ) and Aˆ(Gˆ) be the diagonal matrix of vertex degrees and the mixed adjacency matrix of Gˆ, respectively. The α-mixed adjacency matrix of Gˆ is the matrix Aˆα(Gˆ)=αDˆ(Gˆ)+(1−α)Aˆ(Gˆ).We study some properties of Aˆα(Gˆ) associated with some type of mixed graphs, namely quasi-bipartite and pre-bipartite mixed graphs. A spectral characterization for pre-bipartite and s...
Some families of integral mixed graphs
Andrade, Enide; Bonifácio, Andréa Soares; Robbiano, María; Rodríguez, Jonnathan; Tapia, KatherineA mixed graph $\hat{G}$ is a graph where two vertices can be connected by an edge or by an arc (directed edge). The adjacency matrix , $\hat{A}(\hat{G})$, of a mixed graph has rows and columns indexed by the set of vertices of $\hat{G}$, being its $\{u,v\}$-entry equal to $1$ (respectively, $-1$) if the vertex $u$ is connected by an edge (respectively, an arc) to the vertex $v,$ and $0$ otherwise. These graphs ...
Extremal graphs for Estrada indices
Andrade, Enide; Lenes, Eber; Mallea-Zepeda, Exequiel; Robbiano, María; Rodríguez Z., JonnathanLet $\mathcal{G}$ be a simple undirected connected graph. The signless Laplacian Estrada, Laplacian Estrada and Estrada indices of a graph $\mathcal{G}$ is the sum of the exponentials of the signless Laplacian eigenvalues, Laplacian eigenvalues and eigenvalues of $\mathcal{G}$, respectively. The present work derives an upper bound for the Estrada index of a graph as a function of its chromatic number, in the fa...
On the energy of singular and non singular graphs
Andrade, Enide; Carmona, Juan R.; Poveda, Alex; Robbiano, MaríaLet $G$ be a simple undirected graph with $n$ vertices, $m$ edges, adjacency matrix $A$, largest eigenvalue $\rho$ and nullity $\kappa$. The energy of $G,$ $\mathcal{E}(G)$ is the sum of its singular values. In this work lower bounds for $\mathcal{E}(G)$ in terms of the coefficient of $\mu^{\kappa}$ in the expansion of characteristic polynomial, $p(\mu)=\det{(\mu I-A)}$ are obtained. In particular one of the bo...
A lower bound for the energy of hypoenergetic and non hypoenergetic graphs
Andrade, Enide; R. Carmona, Juan; Infante, Geraldine; Robbiano, MaríaLet $G$ be a simple undirected graph with $n$ vertices and $m$ edges. The energy of $G,$ $\mathcal{E}(G)$ corresponds to the sum of its singular values. This work obtains lower bounds for $\mathcal{E}(G)$ where one of them generalizes a lower bound obtained by Mc Clelland in $1971$ to the case of graphs with given nullity. An extension to the bipartite case is given and, in this case, it is shown that the lower...
On the spectra of some g-circulant matrices and applications to nonnegative inv...
Andrade, Enide; Arrieta, Luis; Manzaneda, Cristina; Robbiano, MaríaA $g$-circulant matrix $A$, is defined as a matrix of order $n$ where the elements of each row of $A$ are identical to those of the previous row, but are moved $g$ positions to the right and wrapped around. Using number theory, certain spectra of $g$-circulant real matrices are given explicitly. The obtained results are applied to Nonnegative Inverse Eigenvalue Problem to construct nonnegative, $g$-circulant ma...
New bounds for the signless Laplacian spread
Andrade, Enide; Dahl, Geir; Leal, Laura; Robbiano, MaríaLet $G$ be an undirected simple graph. The signless Laplacian spread of $G$ is defined as the maximum distance of pairs of its signless Laplacian eigenvalues. This paper establishes some new bounds, both lower and upper, for the signless Laplacian spread. Several of these bounds depend on invariant parameters of the graph. We also use a minmax principle to find several lower bounds for this spectral invariant.
Bounds for different spreads of line and total graphs
Andrade, Enide; Lenes, Eber; Mallea-Zepeda, Exequiel; Robbiano, María; Rodríguez Z., JonnathanIn this paper we explore some results concerning the spread of the line and the total graph of a given graph. A sufficient condition for the spread of a unicyclic graph with an odd girth to be at most the spread of its line graph is presented. Additionally, we derive an upper bound for the spread of the line graph of graphs on $n$ vertices having a vertex (edge) connectivity at most a positive integer $k$. Comb...
New lower bounds for the Randić spread
Andrade, Enide; Freitas, Maria Aguieiras A. de; Robbiano, María; Rodríguez, JonnathanLet $G=\left( \mathcal{V}\left( G\right) ,\mathcal{E}\left( G\right) \right) $ be an $\left( n,m\right) $-graph. The Randi\'{c} spread of $G$, $s_{R}(G)$, is defined as the maximum distance of its Randi\'{c} eigenvalues, disregarding the Randi\'{c} spectral radius of $G$. In this work, we use numerical inequalities and bounds for the matricial spread to obtain relations between this spectral parameter and some ...
Realizable lists on a class of nonnegative matrices
Andrade, Enide; Manzaneda, Cristina; Robbiano, MaríaA square matrix of order $n$ with $n\geq 2$ is called \textit{permutative matrix} when all its rows are permutations of the first row. In this paper recalling spectral results for partitioned into $2$-by-$2$ symmetric blocks matrices sufficient conditions on a given complex list to be the list of the eigenvalues of a nonnegative permutative matrix are given. In particular, we study NIEP and PNIEP when some comp...