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Join the Sons of Liberty! Exploring 1773 Bloxels Boston
No full textLearners will create a colonial character to explore 1773 Boston and gain insight into the underpinnings of the events that led up to and occurred during the Boston Tea Party on December 16, 1773. Using Bloxels, learners will travel the streets of Boston, interacting with different characters and learning about their concerns and events. They will also interact with British soldiers and tax collectors. Additionally, they will be given an opportunity to join the Sons of Liberty and participate in the Boston Tea Party
The allow sequence of distinct eigenvalues for a sign pattern
No full textA sign pattern is a matrix with entries in . This article introduces the allow sequence of distinct eigenvalues for an sign pattern , defined as , with if there exists a real matrix with exactly distinct eigenvalues having pattern , and otherwise. For example, is equivalent to requiring all distinct eigenvalues, while is equivalent to the digraph of being acyclic. Relationships between the allow sequence for and composite cycles of the digraph of are explored to identify zeros in the sequence, while methods based on Jacobian matrices are developed to identify ones in the sequence. When is an irreducible sign pattern, the possible sequences for are completely determined when and when the sequence has at least trailing zeros for
The signless Laplacian spectral radius of graphs without intersecting odd cycles
No full textLet be a graph consisting of cycles of odd length respectively, which intersect in exactly one common vertex, where and . In this paper, we present a sharp upper bound for the signless Laplacian spectral radius of all -free graphs and characterize all extremal graphs which attain the bound. The stability methods and structure of graphs associated with the eigenvalue are adapted for the proof
On invertible non-bipartite unicyclic graphs with a unique perfect matching and their smallest positive eigenvalues
No full textLet be a simple connected graph with the adjacency matrix . By the smallest positive eigenvalue of , we mean the smallest positive eigenvalue of and denote it by . Recently, the smallest positive eigenvalue of bipartite unicyclic graphs with a unique perfect matching has been studied, and the extremal graphs having the minimum and the maximum values have been characterized. We consider the same problem for non-bipartite case. A graph is said to be positively invertible (respectively, negatively invertible) if there exists a signature matrix such that is nonnegative (respectively, nonpositive). In [S. Akbari and S.J. Kirkland. On unimodular graphs. Linear Algebra Appl., 421:3-15, 2007], the authors characterized all the bipartite unicyclic graphs with a unique perfect matching that are positively invertible. In this article, we characterize all the non-bipartite unicyclic graphs with a unique perfect matching that are positively invertible and negatively invertible, respectively. As an application, we obtain the unique graph with the minimum among all the non-bipartite unicyclic graphs on vertices with a unique perfect matching. Except for a specific class, we characterize all other non-bipartite unicyclic graphs with a unique perfect matching such that \tau(G)<\frac{1}{2}. Further, we show that if is a non-bipartite unicyclic graph with a unique perfect matching, then . The extremal graphs with have been obtained. Finally, we obtain the graphs with the maximum among all the non-bipartite unicyclic graphs on vertices with a unique perfect matching
Commuting additive maps on upper triangular and strictly upper triangular infinite matrices
No full textLet be a field, let be the ring of all strictly upper triangular matrices over and let be the ring of all upper triangular matrices over . In this paper, we completely characterize additive maps satisfying for all . As applications, we obtain the finite fields versions of the two main results recently obtained by Slowik and Ahmed [Electron. J. Linear Algebra 37:247-255, 2021]
The numerical range of matrix products
No full textWe discuss what can be said about the numerical range of the matrix product when the numerical ranges of and are known. If two compact convex subsets of the complex plane are given, we discuss the issue of finding a compact convex subset such that whenever () are either unrestricted matrices or normal matrices of the same shape with , it follows that . We do this by defining specific deviation quantities for both the unrestricted case and the normal case
Spectral conditions for rainbow matchings of bipartite graphs
No full textLet be a family of subgraphs of complete bipartite graph for . In this paper, we prove that, if the spectral radius of satisfies for all , then contains a rainbow matching unless
A new weighted spectral geometric mean and properties
No full textIn this paper, we introduce a new weighted spectral geometric mean: \begin{equation*}\label{F-mean}F_t(A,B)= (A^{-1}\sharp_t B)^{1/2} A^{2-2t} (A^{-1} \sharp_t B)^{1/2}, \quad t\in [0,1],\end{equation*} where and are positive definite matrices. We study basic properties and inequalities for . We also establish the Lie-Trotter formula for . Finally, we extend some of the results on to symmetric space of noncompact types
Diameter vs. Laplacian eigenvalue distribution
No full textLet be a simple graph of order . It is known that any Laplacian eigenvalue of belongs to the interval . For an interval , denote by the number of Laplacian eigenvalues of in , counted with multiplicities. Let be the diameter of . If , we show that , and it may be improved into when . We also show that if , and if . The diameter constraint provides an insightful approach to understand how the Laplacian eigenvalues are distributed
