Graphs with few matching roots

We determine all graphs whose matching polynomials have at most five distinct zeros. As a consequence, we find new families of graphs which are determined by their matching polynomial.Comment: 14 pages, 7 figures, 1 appendix table. Final version. Some typos are fixe

Keeping an Open Mind: Challenges and Mysteries in Cancer Cell Biology Research

I received a Summer Undergraduate Research Fellowship (SURF) in 2016 to work in Professor Chuck Walker’s cell biology lab, where I had been involved since 2014. I worked under the guidance of Dr. Walker and alongside other colleagues in the lab. My work during that time focused on culturing cancer cells and investigatingthe proteinp53 within them. The body aims toprevent tumors from forming by protecting the integrity of its cells’ DNA. The protein p53 is so vital in this role that it is often referred to as the “guardian of the genome.” In fact, more than half of all human cancers are associated with malfunctionsthat disrupt p53 function. My project had two objectives. First, I sought to confirm the presence of the p53-mortalin complex in the cells I was planning to use. Second, I tried to disrupt the complex using MKT-077 and withanone and determine the effectiveness of these agents in allowing p53 to move to the nucleus and trigger apoptosis. I planned to designate groups of cells as untreated, MKT-treated, or withanone-treated. For each group, I chose a series of analytical techniques that could pinpoint p53 inthe cell (to see whether it was stuck in the cytoplasm or already in the nucleus) and determine the levels of cell death by apoptosis. The various surprises I experienced while working on my SURF project taught me that research won’t always be as clear-cut as one might expect. I learned the importance of keeping an open mind and considering the possibility of obstacles and unexpected outcomes in order to make sense of conflicting results

Bipartite graphs with five eigenvalues and pseudo designs

A pseudo (v,\, k,\, \la)-design is a pair $(X, {\cal B})$ where $X$ is a $v$-set and ${\cal B}=\{B_1,...,B_{v-1}\}$ is a collection of $k$-subsets (blocks) of $X$ such that each two distinct $B_i, B_j$ intersect in \la elements; and 0\le\la . We use the notion of pseudo designs to characterize graphs of order $n$ whose (adjacency) spectrum contains a zero and $\pm\theta$ with multiplicity $(n-3)/2$ where $0<\theta\le\sqrt{2}$. Meanwhile, partial results confirming a conjecture of O. Marrero on characterization of pseudo (v,\, k,\, \la)-designs are obtained.Comment: 15pages, 6 figures. Final version. To appear in Journal of Algebraic Combinatoric

Spanning trees and even integer eigenvalues of graphs

For a graph $G$, let $L(G)$ and $Q(G)$ be the Laplacian and signless Laplacian matrices of $G$, respectively, and $\tau(G)$ be the number of spanning trees of $G$. We prove that if $G$ has an odd number of vertices and $\tau(G)$ is not divisible by $4$, then (i) $L(G)$ has no even integer eigenvalue, (ii) $Q(G)$ has no integer eigenvalue $\lambda\equiv2\pmod4$, and (iii) $Q(G)$ has at most one eigenvalue $\lambda\equiv0\pmod4$ and such an eigenvalue is simple. As a consequence, we extend previous results by Gutman and Sciriha and by Bapat on the nullity of adjacency matrices of the line graphs. We also show that if $\tau(G)=2^ts$ with $s$ odd, then the multiplicity of any even integer eigenvalue of $Q(G)$ is at most $t+1$. Among other things, we prove that if $L(G)$ or $Q(G)$ has an even integer eigenvalue of multiplicity at least $2$, then $\tau(G)$ is divisible by $4$. As a very special case of this result, a conjecture by Zhou et al. [On the nullity of connected graphs with least eigenvalue at least $-2$, Appl. Anal. Discrete Math. 7 (2013), 250--261] on the nullity of adjacency matrices of the line graphs of unicyclic graphs follows.Comment: Final version. To appear in Discrete Mat