María Robbiano

Universidad Católica del Norte (Chile) · Department of Mathematics

A mixed graph Gˆ is a graph where two vertices can be connected by an edge or by an arc (directed edge). The adjacency matrix, Aˆ(Gˆ), of a mixed graph has rows and columns indexed by the set of vertices of Gˆ, 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 o...

A 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 Eigenvalu...

Let G be a simple undirected connected graph. The signless Laplacian Estrada, Laplacian Estrada and Estrada indices of a graph G is the sum of the exponentials of the signless Laplacian eigenvalues, Laplacian eigenvalues and eigenvalues of G, respectively. The present work derives an upper bound for the Estrada index of a graph as a function of its...

In 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 g-circulant matrix of order n is defined as a matrix of order n where each row is a right cyclic shift in g-places to the preceding row. Using number theory, certain nonnegative g-circulant real matrices are constructed. In particular, it is shown that spectra with sufficient conditions so that it can be the spectrum of a real g-circulant matrix...

Let $R$ be a Hermitian matrix. The energy of $R$, $\mathcal{E}(R)$, corresponds to the sum of the absolute values of its eigenvalues. In this work it is obtained two lower bounds for $\mathcal{E}(R).$ The first one generalizes a lower bound obtained by Mc Clellands for the energy of graphs in $1971$ to the case of Hermitian matrices and graphs with...

Let 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 principl...

In this paper we explore some results concerning the spread of the line and the total graph of a given graph. In particular, it is proved that for an $(n,m)$ connected graph $G$ with $m > n \geq 4$ the spread of $G$ is less than or equal to the spread of its line graph, where the equality holds if and only if $G$ is regular bipartite. A sufficient...

The total graph of G, T (G) is the graph whose vertex set is the union of the sets of vertices and edges of G, where two vertices are adjacent if and only if they stand for either incident or adjacent elements in G. For k ≥ 2, the k-th iterated total graph of G, T k(G), is defined recursively as T k(G) = T (T k-1(G)), where T ¹(G) = T (G) and T ⁰(G...

In this paper we deal with circulant and partitioned into n-by-n circulant blocks matrices and introduce spectral results concerning this class of matrices. The problem of finding lists of complex numbers corresponding to a set of eigenvalues of a nonnegative block matrix with circulant blocks is treated. Along the paper we call realizable list if...

A permutative matrix is a square matrix such that every row is a permutation of the first row. A circulant matrix is a matrix where each row is a cyclic shift of the row above to the right. The Guo's index $\lambda_0$ of a realizable list is the minimum spectral radius such that the list (up to the initial spectral radius) together with $\lambda_0$...

For an undirected simple graph G, the line graph L(G) is the graph whose vertex set is in one-to-one correspondence with the edge set of G where two vertices are adjacent if their corresponding edges in G have a common vertex. The energy E(G) is the sum of the absolute values of the eigenvalues of G. The vertex connectivity κ(G) is referred as the...

The total graph of $G$, $\mathcal T(G)$ is the graph whose set of vertices is the union of the sets of vertices and edges of $G$, where two vertices are adjacent if and only if they stand for either incident or adjacent elements in $G$. Let $\mathcal{T}^1(G)=\mathcal{T}(G)$, the total graph of $G$. For $k\geq2$, the $k\text{-}th$ iterated total gra...

Let $G$ be a 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 fin...

Let $G=\left( V\left( G\right) ,E\left( G\right) \right) $ be an $\left( n,m\right) $-graph and $X$ a nonempty proper subset of $V\left( G\right) $. Let $X^{c}=V\left( G\right) \backslash X$.\ The edge density of $X$ in $G$ is given by \begin{equation*} \rho _{G}\left( X\right) =\frac{n\left\vert E_{X}\left( G\right) \right\vert }{\left\vert X\righ...

For an undirected simple graph G, the line graph L(G) is the graph whose vertex set is in one-to-one correspondence with the edge set of G where two vertices are adjacent if their corresponding edges in G have a common vertex. The energy E(G) is the sum of the absolute values of the eigenvalues of G. The vertex connectivity κ(G) is referred as the...

In this paper we deal with circulant and partitioned into $n$-by-$n$ circulant blocks matrices and introduce spectral results concerning this class of matrices. The problem of finding lists of complex numbers corresponding to a set of eigenvalues of a nonnegative block matrix with circulant blocks is treated. Along the paper we call realizable list...

Let G=(V(G),E(G)) be an (n,m)-graph. The Randić spread of G, sR(G), is defined as the maximum distance of its Randić eigenvalues, disregarding the Randić 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 structural and algebraic paramete...

A square matrix of order n with $n\geq 2$ is called permutative matrix when all its rows (up to the frst one) are permutations of precisely its frst row. In this paper recalling spectral results for partitioned into $2$-by-$2$ symmetric blocks matrices sufcient conditions on a given complex list to be the list of the eigenvalues of a nonnegative pe...

A square matrix of order n with n≥2 is called a permutative matrix or permutative when all its rows (up to the first one) are permutations of precisely its first row. In this paper, the spectra of a class of permutative matrices are studied. In particular, spectral results for matrices partitioned into 2-by-2 symmetric blocks are presented and, usi...

To track the gradual change of the adjacency matrix of a simple graph G into the signless Laplacian matrix, V. Nikiforov in [35] suggested the study of the convex linear combination Aα (α-adjacency matrix),Aα(G)=αD(G)+(1−α)A(G), for α∈[0,1], where A(G) and D(G) are the adjacency and the diagonal vertex degrees matrices of G, respectively. Taking th...

A square matrix of order n with $n \geq 2$ is called permutative matrix when all its rows (up to the frst one) are permutations of precisely its frst row. In this paper, the spectra of a class of permutative matrices are studied. In particular, spectral results for partitioned into 2 by 2 blocks doubly stochastic matrices are presented and, using t...

Let H be an undirected simple graph with vertices v1, . . . , vk and G1, . . . , Gk be a sequence formed with k disjoint graphs Gi, i = 1, . . . , k. The H-generalized composition (or H-join) of this sequence is denoted by H [G1, . . . , Gk] . In this work, we characterize the caterpillar graphs as a H-generalized composition and we study their spe...

The Laplacian Estrada index ( ) and the signless Laplacian Estrada index ( ) of a graph are, respectively, the sum of the exponentials of the eigenvalues of the Laplacian and signless Laplacian matrix of . The vertex frustration index of a graph is the minimum number of vertices whose deletion from results in a bipartite graph. Graphs having maximu...

Some combinatorial and spectral properties of König–Egerváry (K–E) graphs are presented. In particular, some new combinatorial characterizations of K–E graphs are introduced, the Laplacian spectrum of particular families of K–E graphs is deduced, and a lower and upper bound on the largest and smallest adjacency eigenvalue, respectively, of a K–E gr...

The energy of a symmetric matrix is the sum of the absolute values of its eigenvalues. We introduce a lower bound for the energy of a symmetric partitioned matrix into blocks. This bound is related to the spectrum of its quotient matrix. Furthermore, we study necessary conditions for the equality. Applications to the energy of the generalized compo...

The Laplacian spread of a graph G is defined as the difference between the largest and the second smallest eigenvalue of the Laplacian matrix of G. In this work, an upper bound for this graph invariant, that depends on first Zagreb index, is given. Moreover, another upper bound is obtained and expressed as a function of the nonzero coefficients of...

Given a graph G. The fewest number of vertices whose deletion yields a bipartite graph from G was defined by S. Fallat and Yi-Zheng Fan to be the vertex bipartiteness of G and it is denoted by υb(G). We consider the set Σk(n) defined by{G=(V(G),E(G)):G connected,|V(G)|=n and υb(G)≤k}. In this work we identify the graph in Σk(n) with maximum spectra...

The Laplacian-energy-like invariant, LEL, is the sum of the square roots of the Laplacian eigenvalues of the underlying graph G. The incidence energy IE is the sum of the square roots of the signless Laplacian eigenvalues of G. The vertex bipartiteness νb of a graph G is the minimum number of vertices whose deletion from G results in a bipartite gr...

The spread of an $n\times n$ complex matrix $B$ with eigenvalues $\beta _{1},\beta _{2},\ldots ,\beta _{n}$ is defined by \begin{equation*} s\left( B\right) =\max_{i,j}\left\vert \beta _{i}-\beta _{j}\right\vert , \end{equation*}% where the maximum is taken over all pairs of eigenvalues of $B$. Let $G$ be a graph on $n$ vertices. The concept of Lap...

Applying the Cauchy–Schwarz inequality, we obtain a sharp upper bound on the Randić energy of a bipartite graph and of graphs whose adjacency matrix is partitioned into blocks with constant row sum.

Considering a consequence of the Cauchy-Schwarz inequality we obtain a sharp upper bound of the energy of a bipartite graph and a large family graphs, namely those graphs whose adjacency matrix is partitioned into blocks with constant row sum.

Let G be a simple undirected graph of order n with vertex set V(G)={v1,v2,…,vn}V(G)={v1,v2,…,vn}. Let didi be the degree of the vertex vivi. The Randić matrix R=(ri,j)R=(ri,j) of G is the square matrix of order n whose (i,j)(i,j)-entry is equal to 1/didj if the vertices vivi and vjvj are adjacent, and zero otherwise. The Randić energy is the sum of...

Let R be a nonnegative Hermitian matrix. The energy of R, denoted by E(R)E(R), is the sum of absolute values of its eigenvalues. We construct an increasing sequence that converges to the Perron root of R. This sequence yields a decreasing sequence of upper bounds for E(R)E(R). We then apply this result to the Laplacian energy of trees of order n, n...

A new spectral graph invariant $spr_R$, called Randi\'c spread, is defined and investigated. This quantity is equal to the maximal difference between two eigenvalues of the Randi\'c matrix, disregarding the spectral radius. Lower and upper bounds for $spr_R$ are deduced, some of which depending on the Randi\'c index of the underlying graph.

The Randi\'c spread of a simple undirected graph $G$, $spr_\mathbf R(G)$, is equal to the maximal difference between two eigenvalues of the Randi\'c matrix, disregarding the spectral radius [Gomes et al., {\it MATCH Commun. Math. Comput. Chem.\/} {\bf 72} (2014) 249--266]. Using a rank-one perturbation on the Randi\'c matrix of $G$ it is obtained a...

A new spectral graph invariant sprR, called Randic spread, is defined and investigated. This quantity is equal to the maximal difference between two eigenvalues of the Randic matrix, disregarding the spectral radius. Lower and upper bounds for sprR are deduced, some of which depending on the Randic index of the underlying graph.

The spectrum of the Laplacian, signless Laplacian and adjacency matrices of the family of weighted graph $\mathcal{R}\left\{ \mathcal{H}\right\} $ obtained from a connected weighted graph $\mathcal{R}$ on $r$ vertices and $r$ copies of a rooted weighted graph $\mathcal{H}$, by identifying the root of the $i$-th copy of $\mathcal{H}$ with the $i$-th...

The characteristic polynomials of the adjacency matrix of line graphs of caterpillars and then the characteristic polynomials of their Laplacian or signless Laplacian matrices are characterized, using recursive formulas. Furthermore, the obtained results are applied on the determination of upper and lower bounds on the algebraic connectivity of the...

Taking a Fiedler’s result on the spectrum of a matrix formed from two symmetric matrices as a motivation, a more general result is deduced and applied to the determination of adjacency and Laplacian spectra of graphs obtained by a generalized join graph operation on families of graphs (regular in the case of adjacency spectra and arbitrary in the c...

In this paper we give a simple characterization of the Laplacian spectra of a family of graphs as the eigenvalues of symmetric tridiagonal matrices. In addition, we apply our result to obtain upper and lower bounds for the Laplacian-energy-like invariant of these graphs. The class of graphs considered are obtained from copies of modified generalize...

Consider the Laplacian and signless Laplacian spectrum of a graph G of order n, with k pairwise co-neighbor vertices. We prove that the number of shared neighbors is a Laplacian and a signless Laplacian eigenvalue of G with multiplicity at least k − 1. Additionally, considering a connected graph GkGk with a vertex set defined by the k pairwise co-n...

Considering a graph $H$ of order $p$, a generalized $H$-join operation of a family of graphs $G_1,..., G_p$, constrained by a family of vertex subsets $S_i \subseteq V(G_i)$, $i=1,..., p,$ is introduced. When each vertex subset $S_i$ is $(k_i,\tau_i)$-regular, it is deduced that all non-main adjacency eigenvalues of $G_i$, different from $k_i-\tau_...

The energy of a graph G is the sum of the absolute values of the eigenvalues of the adjacency matrix of G. The Laplacian (respectively, the signless Laplacian) energy of G is the sum of the absolute values of the differences between the eigenvalues of the Laplacian (respectively, signless Laplacian) matrix and the arithmetic mean of the vertex degr...

In a previous paper [M. Robbiano, E.A. Martins, and I. Gutman, Extending a theorem by Fiedler and applications to graph energy, MATCH Commun. Math. Comput. Chem. 64 (2010), pp. 145–156], a lemma by Fiedler was used to obtain eigenspaces of graphs, and applied to graph energy. In this article Fiedler's lemma is generalized and this generalization is...

Let B be a weighted generalized Bethe tree of k levels (k > 1) in which nj is the number of vertices at the level k-j+1 (1 ≤ j ≤ k). Let Δ ⊆ {1, 2,., k-1} and F={Gj:j∈δ}, where Gj is a prescribed weighted graph on each set of children of B at the level k-j+1. In this paper, the eigenvalues of a block symmetric tridiagonal matrix of order n1+n2 +......

The energy of a graph is equal to the sum of the absolute values of its eigenvalues. The energy of a matrix is equal to the sum of its singular values. We establish relations between the energy of the line graph of a graph G and the energies associated with the Laplacian and signless Laplacian matrices of G.

A Bethe tree of k levels, Bk(d), is a rooted tree such that the root vertex has degree d, the vertices from level 2 to k−1 have degree d+1 and the vertices at level k are leaves. In this paper, we obtain a recurrence relation for the characteristic polynomial and the Laplacian characteristic polynomial of Bethe trees. As an application, we prove th...

For k,d⩾2, a Bethe tree is a rooted tree with k levels which the root vertex has degree d, the vertices from level 2 to k-1 have degree d+1 and the vertices at the level k are pendent vertices. So et al [W. So, M. Robbiano, N. M. M. de Abreu, and I. Gutman, “Applications of a theorem by Ky Fan in the theory of graph energy,” Linear Algebra Appl. 43...

The energy of a graph G is equal to the sum of the absolute values of the eigenvalues of G, which in turn is equal to the sum of the singular values of the adjacency matrix of G. Let X, Y, and Z be matrices, such that X+Y=Z. The Ky Fan theorem establishes an inequality between the sum of the singular values of Z and the sum of the sum of the singul...

The Laplacian energy LE[G] of a simple graph G with n vertices and m edges is equal to the sum of distances of the Laplacian eigenvalues to their average. For 1 <= j <= s, let A(j) be matrices of orders n(j). Suppose that det(L(G) - lambda I(n)) = Pi(s)(j=1) det(A(j) - lambda I(nj))(tj), with tj > 0. In the present paper we prove LE[G] <= Sigma(s)(...

We use a lemma due to Fiedler to obtain eigenspaces of some graphs and apply these results to graph energy (= the sum of absolute values of the graph eigenvalues = the sum of singular values of the adjacency matrix). We obtain some new upper and lower bounds for graph energy and find new examples of graphs whose energy exceeds the number of vertice...

The energy of a graph $G$ is the sum of the absolute values of the eigenvalues of the adjacency matrix of $G$. The Laplacian (respectively, the signless Laplacian) energy of $G$ is the sum of the absolute values of the differences between the eigenvalues of the Laplacian (respectively, signless Laplacian) matrix and the arithmetic mean of the verte...

The Bethe trees are unweighted rooted trees of k levels whose root vertex has degree d, the vertices from 2 to level k -1 have degree d + 1 and the vertices at level k have degree 1. This paper gives an explicit formula for the energy and an approximation to the Estrada index of Bethe trees. We also obtain an explicit formula for the energy and an...

The Laplacian energy of a graph G is equal to the sum of distances of the Laplacian eigenvalues from their average, which in turn is equal to the sum of singular values of a shift of Laplacian matrix of G. Let X,Y and Z be matrices, such that Z=X+Y. Ky Fan has established an inequality between the sum of singular values of Z and the sum of the sum...

The Bethe tree Bd,k is the rooted tree of k levels whose root vertex has degree d, the vertices from level 2 to level k -1 have degree d+1, and the vertices at level k have degree 1. This paper gives a decomposition of the characteristic polynomial of the adjacency matrix of the tree T(d, k, r), obtained by attaching copies of B(d, k) to the vertic...

A Bethe tree Bd,k is a rooted unweighted of k levels in which the root vertex has degree equal to d, the vertices at level j(2⩽j⩽k-1) have degree equal to (d+1) and the vertices at level k are the pendant vertices. In this paper, we first derive an explicit formula for the eigenvalues of the adjacency matrix of Bd,k. Moreover, we give the correspon...

Let T be a weighted rooted tree of k levels such that(1)the vertices in level j have a degree equal to dk−j+1 for j=1,2,…,k, and(2)the edges joining the vertices in level j with the vertices in level (j+1) have a weight equal to wk−j for j=1,2,…,k−1.We give a complete characterization of the eigenvalues of the Laplacian matrix and adjacency matrix...