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Let G be a graph. The energy of G, E (G) , is defined as the sum of absolute values of its eigenvalues. Here, it is shown that if G is a graph and { H 1 , ... , H k } is an edge partition of G, such that H 1 , ... , H k are spanning; then E (G) = ∑ i = 1 k E (H i) if and only if A i A j = 0 , for every 1 ⩽ i , j ⩽ k and i ≠ j , where A i is the adjacency matrix of H i . It was proved that if G is a graph and H 1 , ... , H k are subgraphs of G which partition edges of G, then E (G) ⩽ ∑ i = 1 k E (H i). In this paper we show that if G is connected, then the equality is strict, that is E (G) < ∑ i = 1 k E (H i). [ABSTRACT FROM AUTHOR]

In this paper, we propose a modified generalized SOR-like (MGSOR) method for solving an absolute value equation (AVE), which is obtained by reformulating equivalently AVE as a two-by-two block nonlinear equation and by introducing the transformation P y := | x | with a general nonsingular matrix P. The convergence results of the MGSOR method are obtained under certain assumptions imposed on the involved parameters. Furthermore, the optimal parameters minimizing the convergence rate of the MGSOR method for solving AVE are studied in detail. Numerical experiments further illustrate that the MGSOR method is efficient and has better performance than some existing iteration methods in aspects of the number of iteration steps and CPU time. [ABSTRACT FROM AUTHOR]

Two graphs having the same spectrum are said to be cospectral. A pair of singularly cospectral graphs is formed by two graphs such that the absolute values of their nonzero eigenvalues coincide. Clearly, a pair of cospectral graphs is also singularly cospectral but the converse may not be true. Two graphs are almost cospectral if their nonzero eigenvalues and their multiplicities coincide. In this paper, we present necessary and sufficient conditions for a pair of graphs to be singularly cospectral, giving an answer to a problem posted by Nikiforov. In addition, we construct an infinite family of pairs of noncospectral singularly cospectral graphs with unbounded number of vertices. It is clear that almost cospectral graphs are also singularly cospectral but the converse is not necessarily true, we present families of graphs where both concepts: almost cospectrality and singularly cospectrality agree. [ABSTRACT FROM AUTHOR]

The eccentricity matrix ϵ (G) of a connected graph G is obtained from the distance matrix of G by retaining the largest distance in each row and each column and setting the remaining entries as 0. The eccentricity matrices of graphs are closely related to the distance matrices of graphs, nevertheless a number of properties of eccentricity matrices are substantially different from those of the distance matrices. The eccentricity matrices of trees need not be invertible though the distance matrices of trees are invertible. In this paper, we establish a necessary and sufficient condition for the eccentricity matrix of a tree to be invertible. The largest eigenvalue of ϵ (G) is called the ε-spectral radius, and the eccentricity energy (or the ε-energy) of G is the sum of the absolute values of the eigenvalues of ϵ (G). We obtain some bounds for the ε-spectral radius and characterize the extreme graphs. Two graphs are said to be ε-equienergetic if they have the same ε-energy. For any n ≥ 5 , we construct a pair of ε-equienergetic graphs on n vertices, which are not ε-cospectral. [ABSTRACT FROM AUTHOR]

The energy of a graph is defined as the sum of the absolute values of all eigenvalues with respect to its adjacency matrix. Denote by Gn,m the set of all connected graphs having n vertices and m edges. Caporossi et al. [Caporossi G, Cvetkovi D, Gutman I, et al. Variable neighbourhood search for extremal graphs. 2. Finding graphs with external energy. J Chem Inf Comput Sci. 1999;39:984–996] conjectured that among all graphs in Gn,m, n ≥ 6 and n − 1 ≤ m ≤ 2(n − 2), the graphs with minimum energy are the star Sn with m−n + 1 additional edges all connected to the same vertices for m ≤ n + ⌊ (n − 7) / 2 ⌋ , and the bipartite graph with two vertices on one side, one of which is connected to all vertices on the other side, otherwise. In this paper, we provide a new approach to investigate the conjecture above. Especially, we determine the unique tetracyclic graph having minimum energy. [ABSTRACT FROM AUTHOR]

Let S(G) be the Seidel matrix of a graph G. The Seidel energy of G, denoted by ES(G), is defined to be the sum of absolute values of all eigenvalues of the Seidel matrix S(G) of G. In this paper, we consider the change of Seidel energy of graphs due to edge deletion. It is proved that the Seidel energy of the tripartite Turán graph T(n, 3) of order n ≥ 9 is always increased when an edge is deleted. [ABSTRACT FROM AUTHOR]

Let G be a graph and S (G) be the Seidel matrix of G. The Seidel energy E s (G) is defined to be the sum of the absolute value of all eigenvalues of the Seidel matrix of G. In this paper, we study some properties of the Seidel eigenvalues of G. Moreover, we present some lower and upper bounds for the Seidel energy of G. [ABSTRACT FROM AUTHOR]

The energy E (G) of a graph G is the sum of the absolute values of all eigenvalues of G. More recently, the authors of Wong, Wang and Chu [Lower bounds of graph energy in terms of matching number. Linear Algebra Appl. 2018;549:276–286] obtained a lower bound for graph energy. They proved that, for a connected graph G apart from K 1 , K 3 and K p , q with 1 ≤ p q ≤ 4 , the energy of G is at least 2 5 . In the present paper, we aim to improve the lower bound from 2 5 to 6. The extreme graphs with energy 6 are characterized. [ABSTRACT FROM AUTHOR]

In this paper, we study spectral properties of complex weighted digraphs. We show that a complex weighted digraph D is balanced if and only if D and | D | have the same spectrum, where | D | is the absolute value weighted digraph of D, that is, the digraph obtained by replacing the weight of each arc by its absolute value. We extend the concept of real energy to complex weighted digraphs and obtain extremal energy unicyclic complex weighted digraphs with cycle-weight in the punctured disk Δ = { z ∈ C : | z | ≤ 1 } ∖ { 0 }. We consider a family of complex weighted digraphs D n , h , in which each digraph has order n and cycles of length h ≥ 2 only with constant complex weight c = a + ι ˙ b. We show that for each D ∈ D n , h , the real energy of D is related to the real energy of unweighted cycle of length h and in some special cases real energy can be compared using quasi-order relation on coefficients of the characteristic polynomial. Finally, we obtain upper bounds on the real energy which generalize those known for unweighted digraphs and signed digraphs. [ABSTRACT FROM AUTHOR]

For every real matrix A having as an eigenvector e, the all-ones vector, an upper bound is obtained for the absolute value of its eigenvalues except, in some cases, the one associated with e. Comparisons are made between this bound and some existing bounds for stochastic matrices as well as other types of matrices such as Laplacian matrices and Randić matrices. Other examples and results are also provided. An interesting recent article 'An eigenvalue localization theorem for stochastic matrices and its application to Randić matrices', Linear Algebra Appl. 2016;505:85-96, by A. Banerjee and R. Mehatari, is useful for this present paper. [ABSTRACT FROM AUTHOR]