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The absorption laws for the generalized inverses in rings
No full textIn this paper, it is given equivalent conditions for the absorption laws in terms of the Moore-Penrose, group, core inverse, core inverse dual, {1}, {1,2}, {1,3}, and {1,4} inverses in rings. The results given here extend the results of [X. Liu, H. Jin, and D.S. Cvetkovi´c-Ili´c. The absorption laws for the generalized inverses. Appl. Math. Comp., 219:2053â2059, 2012]
A Lagrange series approach to the spectrum of the Kite
No full textA Lagrange series around adjustable expansion points to compute the eigenvalues of graphs, whose characteristic polynomial is analytically known, is presented. The computations for the kite graph P_nK_m, whose largest eigenvalue was studied by Stevanovic and Hansen [D. Stevanovic and P. Hansen. The minimum spectral radius of graphs with a given clique number. Electronic Journal of Linear Algebra, 17:110â117, 2008.], are illustrated. It is found that the first term in the Lagrange series already leads to a better approximation than previously published bounds
Godsil-McKay switching and isomorphism
No full textGodsil-McKay switching is an operation on graphs that doesnât change the spectrum of the adjacency matrix. Usually (but not always) the obtained graph is non-isomorphic with the original graph. We present a straightforward sufficient condition for being isomorphic after switching, and give examples which show that this condition is not necessary. For some graph products we obtain sufficient conditions for being non-isomorphic after switching. As an example we find that the tensor product of the grid L(â,m) (â > m>2) and a graph with at least one vertex of degree two is not determined by its adjacency spectrum
Graphs with many valencies and few eigenvalues
No full textDom de Caen posed the question whether connected graphs with three distinct eigenvalues have at most three distinct valencies. We do not answer this question, but instead construct connected graphs with four and five distinct eigenvalues and arbitrarily many distinct valencies. The graphs with four distinct eigenvalues come from regular two-graphs. As a side result, we characterize the disconnected graphs and the graphs with three distinct eigenvalues in the switching class of a regular two-graph
Bounding the CP-rank by graph parameters
No full textThe cp-rank of a graph G, cpr(G), is the maximum cp-rank of a completely positive matrix with graph G. One obvious lower bound on cpr(G) is the (edge-) clique covering number, cc(G), i.e., the minimal number of cliques needed to cover all of Gâs edges. It is shown here that for a connected graph G, cpr(G) = cc(G) if and only if G is triangle free and not a tree. Another lower bound for cpr(G) is tf(G), the maximum size of a triangle free subgraph of G. We consider the question of when does the equality cpr(G) = tf(G) hold
On the Chudnovsky-Seymour-Sullivan conjecture on cycles in triangle-free digraphs
No full textFor a simple digraph G without directed triangles or digons, let \beta(G) be the size of the smallest subset X of E(G) such that G\X has no directed cycles, and let \gamma(G) be the number of unordered pairs of nonadjacent vertices in G. In 2008, Chudnovsky, Seymour, and Sullivan showed that \beta(G) \leq \gamma(G) and conjectured that \beta(G) \leq \gamma(G)/2. Recently, Dunkum, Hamburger, and Por proved that \beta(G)\leq .88\gamma(G). In this note, we prove that \beta(G) \leq .8616 \gamma(G)
Preface: Special volume of Electronic Journal of Linear Algebra dedicated to Professor Ravindra B. Bapat
No full textThis special volume of the Electronic Journal of Linear Algebra is dedicated to Professor Ravindra B. Bapat on the occasion of his 60th birthday. The volume contains papers related to the International Conference on Linear Algebra & its Applications, which was held December 18--20, 2014 at the Department of Statistics of Manipal University in Manipal, India. The theme of conference focused on (i) Matrix Methods in Statistics, (ii) Combinatorial Matrix Theory and (iii) Classical Matrix Theory covering different aspects of Linear Algebra
Finite and infinite structures of rational matrices: a local approach
No full textThe structure of a rational matrix is given by its Smith-McMillan invariants. Some properties of the Smith-McMillan invariants of rational matrices with elements in different principal ideal domains are presented: In the ring of polynomials in one indeterminate (global structure), in the local ring at an irreducible polynomial (local structure), and in the ring of proper rational functions (infinite structure). Furthermore, the change of the finite (global and local) and infinite structures is studied when performing a Mobius transformation on a rational matrix. The results are applied to define an equivalence relation in the set of polynomial matrices, with no restriction on size, for which a complete system of invariants are the finite and infinite elementary divisors
Properties of first eigenvectors and eigenvalues of nonsingular weighted directed graphs
No full textThe class of nonsingular connected weighted directed graphs with an unweighted undirected branch is considered in this article. This paper investigates the monotonicity properties of the first eigenvector of such graphs along certain paths. The paper describes how the first eigenvalue of such graphs changes under some perturbation. It is shown that replacing a branch which is a tree by a path on the same number of vertices will not increase the first eigenvalue, while replacing the tree by a star on the same number of vertices will not decrease the first eigenvalue. As an application the paper characterizes the graphs minimizing the first eigenvalue over certain classes of such graphs
On the Robust Stability of Polynomial Matrix Families
No full textIn this study, the problem of robust asymptotic stability of n by n polynomial matrix family, in both continuous-time and discrete-time cases, is considered. It is shown that in the continuous case the problem can be reduced to positivity of two specially constructed multivariable polynomials, whereas in the discrete-time case it is required three polynomials. A number of examples are given, where the Bernstein expansion method and sufficient conditions from [L.H. Keel and S.P. Bhattacharya. Robust stability via sign-definite decomposition. IEEE Transactions on Automatic Control, 56(1):140â145, 2011.] are applied to test positivity of the obtained multivariable polynomials. Sufficient conditions for matrix polytopes and one interesting negative result for companion matrices are also considered