Let G be a graph on n vertices and λ 1 ≥ λ 2 ≥. .. ≥ λ n its eigenvalues. The Estrada index of G ... more Let G be a graph on n vertices and λ 1 ≥ λ 2 ≥. .. ≥ λ n its eigenvalues. The Estrada index of G is defined as EE(G) = n i=1 e λ i. In this work, we use an increasing sequence converging to the λ 1 to obtain an increasing sequence of lower bounds for EE(G). In addition, we generalize this succession for the Estrada index of an arbitrary symmetric nonnegative matrix.
Let $G$ be a graph on $n$ vertices and $\lambda_1,\lambda_2,\ldots,\lambda_n$ its eigenvalues. Th... more Let $G$ be a graph on $n$ vertices and $\lambda_1,\lambda_2,\ldots,\lambda_n$ its eigenvalues. The Estrada index of $G$ is defined as $EE(G)=\sum_{i=1}^n e^{\lambda_i}.$ In this work, using a different demonstration technique, new lower bounds are obtained for the Estrada index, that depends on the number of vertices, the number of edges and the energy of the graph is given. Moreover, another lower bound for the Estrada index is obtained of an arbitrary non-negative Hermitian matrix are established.
Let G be a graph on n vertices and λ1, λ2, . . . , λn its eigenvalues. The Estrada index of G is ... more Let G be a graph on n vertices and λ1, λ2, . . . , λn its eigenvalues. The Estrada index of G is defined as EE(G) = ∑n i=1 e λi . In this work, using a different demonstration technique, new lower bounds are obtained for the Estrada index, that depends on the number of vertices, the number of edges and the energy of the graph is given. Moreover, another lower bound for the Estrada index is obtained of an arbitrary Hermitian matrix are established.
In this paper, we postulate a new decomposition theorem of a matrix A into two matrices, namely, ... more In this paper, we postulate a new decomposition theorem of a matrix A into two matrices, namely, a lower triangular matrix M, in which all entries are determinants, and an upper triangular matrix U whose entries are also in determinant form. From a well-known theorem on the pivot elements of the Doolittle-Gauss elimination process, we deduce a corollary to obtain a diagonal matrix D. With it, we scale the elementary lower triangular matrix of the Doolittle- Gauss elimination process and deduce a new elementary lower triangular matrix. Applying this linear transformation to A by means of both minimum and complete pivoting strategies, we obtain the determinant of A as if it had been calculated by means of a Laplace expansion. If we apply this new linear transformation and the above pivot strategy to an augmented matrix (A|b), we obtain a Cramer's solution of the linear
Let $R$ be a Hermitian matrix. The energy of $R$, $\mathcal{E}(R)$, corresponds to the sum of the... more 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 a given nullity. The second one generalizes a lower bound obtained by K. Das, S. A. Mojallal and I. Gutman in 2013 to symmetric non-negative matrices and graphs with a given nullity. The equality cases are discussed. These lower bounds are obtained for graphs with $m$ edges and some examples are provided showing that, some obtained bounds are incomparable with the known lower bound for the energy $2\sqrt{m}$. Another family of lower bounds are obtained from an increasing sequence of lower bounds for the spectral radius of a graph. The bounds are stated for singular and non-singular graphs.
Let $D$ be a simple digraph with eigenvalues $z_1,z_2,...,z_n$. The energy of $D$ is defined as $... more Let $D$ be a simple digraph with eigenvalues $z_1,z_2,...,z_n$. The energy of $D$ is defined as $E(D)= \sum_{i=1}^n |Re(z_i)|$, is the real part of the eigenvalue $z_i$. In this paper a lower bound will be obtained for the spectral radius of $D$, wich improves some the lower bounds that appear in the literature \cite{G-R}, \cite{T-C}. This result allows us to obtain an upper bound for the energy of $ D $. Finally, digraphs are characterized in which this upper bound improves the bounds given in \cite{G-R} and \cite{T-C}.
Let $G$ be a graph on $n$ vertices and $\lambda_1,\lambda_2,\ldots,\lambda_n$ its eigenvalues. Th... more Let $G$ be a graph on $n$ vertices and $\lambda_1,\lambda_2,\ldots,\lambda_n$ its eigenvalues. The Estrada index of $G$ is defined as $EE(G)=\sum_{i=1}^n e^{\lambda_i}.$ In this work, using a different demonstration technique, new lower bounds are obtained for the Estrada index, that depends on the number of vertices, the number of edges and the energy of the graph is given. Moreover, another lower bound for the Estrada index is obtained of an arbitrary non-negative Hermitian matrix are established.
Match-communications in Mathematical and in Computer Chemistry, 2020
Let G be a simple undirected graph with n vertices and m edges. The energy of G, E(G) corresponds... more Let G be a simple undirected graph with n vertices and m edges. The energy of G, E(G) corresponds to the sum of its singular values. This work obtains lower bounds for 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 bound 2 √ m is improved. The equality cases are characterized. Moreover, a simple lower bound that considers the number of edges and the diameter of G is derived. A simple lower bound, which improves the lower bound 2 √ n− 1, for the energy of trees with n vertices and diameter d is also obtained. 1 Notation and Preliminaries In this work we deal with an (n,m)-graph G which is an undirected simple graph with vertex set V (G) and edge set E (G) of cardinality n and m, respectively. As usual we denote the adjacency matrix of G by A = A(G). The eigenvalues of G are the eigenvalues of A (see e.g. [5, 6]). Its eig...
Match-communications in Mathematical and in Computer Chemistry, 2020
Let G be a simple undirected graph with n vertices, m edges, adjacency matrix A, largest eigenval... more Let G be a simple undirected graph with n vertices, m edges, adjacency matrix A, largest eigenvalue ρ and nullity κ. The energy of G, E(G) is the sum of its singular values. In this work lower bounds for E(G) in terms of the coefficient of μκ in the expansion of characteristic polynomial, p(μ) = det (μI −A) are obtained. In particular one of the bounds generalizes a lower bound obtained by K. Das, S. A. Mojallal and I. Gutman in 2013 to the case of graphs with given nullity. The bipartite case is also studied obtaining in this case, a sufficient condition to improve the spectral lower bound 2ρ. Considering an increasing sequence convergent to ρ a convergent increasing sequence of lower bounds for the energy of G is constructed.
Let G be a graph on n vertices and λ 1 , λ 2 ,. .. , λ n its eigenvalues. The Estrada index of G ... more Let G be a graph on n vertices and λ 1 , λ 2 ,. .. , λ n its eigenvalues. The Estrada index of G is an invariant that is calculated from the eigenvalues of the adjacency matrix of a graph. In this paper, we present some new lower bounds obtained for the Estrada Index of graphs and in particular of bipartite graphs that only depend on the number of vertices, the number of edges, Randić index, maximum and minimum degree and diameter.
Let G be a graph on n vertices and λ 1 ≥ λ 2 ≥. .. ≥ λ n its eigenvalues. The Estrada index of G ... more Let G be a graph on n vertices and λ 1 ≥ λ 2 ≥. .. ≥ λ n its eigenvalues. The Estrada index of G is defined as EE(G) = n i=1 e λ i. In this work, we use an increasing sequence converging to the λ 1 to obtain an increasing sequence of lower bounds for EE(G). In addition, we generalize this succession for the Estrada index of an arbitrary symmetric nonnegative matrix.
Let G be a graph on n vertices and λ 1 ≥ λ 2 ≥. .. ≥ λ n its eigenvalues. The Estrada index of G ... more Let G be a graph on n vertices and λ 1 ≥ λ 2 ≥. .. ≥ λ n its eigenvalues. The Estrada index of G is defined as EE(G) = n i=1 e λ i. In this work, we use an increasing sequence converging to the λ 1 to obtain an increasing sequence of lower bounds for EE(G). In addition, we generalize this succession for the Estrada index of an arbitrary symmetric nonnegative matrix.
Let $G$ be a graph on $n$ vertices and $\lambda_1,\lambda_2,\ldots,\lambda_n$ its eigenvalues. Th... more Let $G$ be a graph on $n$ vertices and $\lambda_1,\lambda_2,\ldots,\lambda_n$ its eigenvalues. The Estrada index of $G$ is defined as $EE(G)=\sum_{i=1}^n e^{\lambda_i}.$ In this work, using a different demonstration technique, new lower bounds are obtained for the Estrada index, that depends on the number of vertices, the number of edges and the energy of the graph is given. Moreover, another lower bound for the Estrada index is obtained of an arbitrary non-negative Hermitian matrix are established.
Let G be a graph on n vertices and λ1, λ2, . . . , λn its eigenvalues. The Estrada index of G is ... more Let G be a graph on n vertices and λ1, λ2, . . . , λn its eigenvalues. The Estrada index of G is defined as EE(G) = ∑n i=1 e λi . In this work, using a different demonstration technique, new lower bounds are obtained for the Estrada index, that depends on the number of vertices, the number of edges and the energy of the graph is given. Moreover, another lower bound for the Estrada index is obtained of an arbitrary Hermitian matrix are established.
In this paper, we postulate a new decomposition theorem of a matrix A into two matrices, namely, ... more In this paper, we postulate a new decomposition theorem of a matrix A into two matrices, namely, a lower triangular matrix M, in which all entries are determinants, and an upper triangular matrix U whose entries are also in determinant form. From a well-known theorem on the pivot elements of the Doolittle-Gauss elimination process, we deduce a corollary to obtain a diagonal matrix D. With it, we scale the elementary lower triangular matrix of the Doolittle- Gauss elimination process and deduce a new elementary lower triangular matrix. Applying this linear transformation to A by means of both minimum and complete pivoting strategies, we obtain the determinant of A as if it had been calculated by means of a Laplace expansion. If we apply this new linear transformation and the above pivot strategy to an augmented matrix (A|b), we obtain a Cramer's solution of the linear
Let $R$ be a Hermitian matrix. The energy of $R$, $\mathcal{E}(R)$, corresponds to the sum of the... more 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 a given nullity. The second one generalizes a lower bound obtained by K. Das, S. A. Mojallal and I. Gutman in 2013 to symmetric non-negative matrices and graphs with a given nullity. The equality cases are discussed. These lower bounds are obtained for graphs with $m$ edges and some examples are provided showing that, some obtained bounds are incomparable with the known lower bound for the energy $2\sqrt{m}$. Another family of lower bounds are obtained from an increasing sequence of lower bounds for the spectral radius of a graph. The bounds are stated for singular and non-singular graphs.
Let $D$ be a simple digraph with eigenvalues $z_1,z_2,...,z_n$. The energy of $D$ is defined as $... more Let $D$ be a simple digraph with eigenvalues $z_1,z_2,...,z_n$. The energy of $D$ is defined as $E(D)= \sum_{i=1}^n |Re(z_i)|$, is the real part of the eigenvalue $z_i$. In this paper a lower bound will be obtained for the spectral radius of $D$, wich improves some the lower bounds that appear in the literature \cite{G-R}, \cite{T-C}. This result allows us to obtain an upper bound for the energy of $ D $. Finally, digraphs are characterized in which this upper bound improves the bounds given in \cite{G-R} and \cite{T-C}.
Let $G$ be a graph on $n$ vertices and $\lambda_1,\lambda_2,\ldots,\lambda_n$ its eigenvalues. Th... more Let $G$ be a graph on $n$ vertices and $\lambda_1,\lambda_2,\ldots,\lambda_n$ its eigenvalues. The Estrada index of $G$ is defined as $EE(G)=\sum_{i=1}^n e^{\lambda_i}.$ In this work, using a different demonstration technique, new lower bounds are obtained for the Estrada index, that depends on the number of vertices, the number of edges and the energy of the graph is given. Moreover, another lower bound for the Estrada index is obtained of an arbitrary non-negative Hermitian matrix are established.
Match-communications in Mathematical and in Computer Chemistry, 2020
Let G be a simple undirected graph with n vertices and m edges. The energy of G, E(G) corresponds... more Let G be a simple undirected graph with n vertices and m edges. The energy of G, E(G) corresponds to the sum of its singular values. This work obtains lower bounds for 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 bound 2 √ m is improved. The equality cases are characterized. Moreover, a simple lower bound that considers the number of edges and the diameter of G is derived. A simple lower bound, which improves the lower bound 2 √ n− 1, for the energy of trees with n vertices and diameter d is also obtained. 1 Notation and Preliminaries In this work we deal with an (n,m)-graph G which is an undirected simple graph with vertex set V (G) and edge set E (G) of cardinality n and m, respectively. As usual we denote the adjacency matrix of G by A = A(G). The eigenvalues of G are the eigenvalues of A (see e.g. [5, 6]). Its eig...
Match-communications in Mathematical and in Computer Chemistry, 2020
Let G be a simple undirected graph with n vertices, m edges, adjacency matrix A, largest eigenval... more Let G be a simple undirected graph with n vertices, m edges, adjacency matrix A, largest eigenvalue ρ and nullity κ. The energy of G, E(G) is the sum of its singular values. In this work lower bounds for E(G) in terms of the coefficient of μκ in the expansion of characteristic polynomial, p(μ) = det (μI −A) are obtained. In particular one of the bounds generalizes a lower bound obtained by K. Das, S. A. Mojallal and I. Gutman in 2013 to the case of graphs with given nullity. The bipartite case is also studied obtaining in this case, a sufficient condition to improve the spectral lower bound 2ρ. Considering an increasing sequence convergent to ρ a convergent increasing sequence of lower bounds for the energy of G is constructed.
Let G be a graph on n vertices and λ 1 , λ 2 ,. .. , λ n its eigenvalues. The Estrada index of G ... more Let G be a graph on n vertices and λ 1 , λ 2 ,. .. , λ n its eigenvalues. The Estrada index of G is an invariant that is calculated from the eigenvalues of the adjacency matrix of a graph. In this paper, we present some new lower bounds obtained for the Estrada Index of graphs and in particular of bipartite graphs that only depend on the number of vertices, the number of edges, Randić index, maximum and minimum degree and diameter.
Let G be a graph on n vertices and λ 1 ≥ λ 2 ≥. .. ≥ λ n its eigenvalues. The Estrada index of G ... more Let G be a graph on n vertices and λ 1 ≥ λ 2 ≥. .. ≥ λ n its eigenvalues. The Estrada index of G is defined as EE(G) = n i=1 e λ i. In this work, we use an increasing sequence converging to the λ 1 to obtain an increasing sequence of lower bounds for EE(G). In addition, we generalize this succession for the Estrada index of an arbitrary symmetric nonnegative matrix.
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