Unicyclic Graphs with equal Laplacian Energy

We introduce a new operation on a class of graphs with the property that the Laplacian eigenvalues of the input and output graphs are related. Based on this operation, we obtain a family of order (square root of n) noncospectral unicyclic graphs on n vertices with the same Laplacian energy.Comment: 11 pages, 11 figures, slightly modified version of Theorem 1 when compared with original pape

Euclid’s Elements for High School Classrooms

What is our goal when teaching students geometry? Is it to obtain a passing grade, be able to construct geometric figures, know all of the necessary terms? I would like to propose that the purpose is to cultivate a love for the logic, art, and argument of geometry. The original geometry book written by Euclid was used as a guide for two thousand years and led many of our historical geometers to discoveries. Instead of relying on memorizing confusing acronyms for congruent figures, let\u27s give the students an opportunity to see the work behind the properties. By using the methods outlined in this curriculum, students will experience geometry, study how to accurately justify their work, learn how to discuss, disagree, and defend respectfully, and know the math behind the theorems. To evaluate the curriculum, a small pilot study was conducted on the impact on student learning. Moreover, the impact and value of this curriculum was evaluated by two experts who specialize in teaching mathematics at the high school level

Limits of permutation sequences

A permutation sequence is said to be convergent if the density of occurrences of every fixed permutation in the elements of the sequence converges. We prove that such a convergent sequence has a natural limit object, namely a Lebesgue measurable function $Z:[0,1]^2 \to [0,1]$ with the additional properties that, for every fixed $x \in [0,1]$, the restriction $Z(x,\cdot)$ is a cumulative distribution function and, for every $y \in [0,1]$, the restriction $Z(\cdot,y)$ satisfies a "mass" condition. This limit process is well-behaved: every function in the class of limit objects is a limit of some permutation sequence, and two of these functions are limits of the same sequence if and only if they are equal almost everywhere. An ingredient in the proofs is a new model of random permutations, which generalizes previous models and might be interesting for its own sake.Comment: accepted for publication in the Journal of Combinatorial Theory, Series B. arXiv admin note: text overlap with arXiv:1106.166