Some Applications of Wagner's Weighted Subgraph Counting Polynomial

We use Wagner's weighted subgraph counting polynomial to prove that the partition function of the anti-ferromagnetic Ising model on line graphs is real rooted and to prove that roots of the edge cover polynomial have absolute value at most $4$. We more generally show that roots of the edge cover polynomial of a $k$-uniform hypergraph have absolute value at most $2^k$, and discuss applications of this to the roots of domination polynomials of graphs. We moreover discuss how our results relate to efficient algorithms for approximately computing evaluations of these polynomials.

On Edge Covering Chain of a Graph

Abstract- In the study of domination in graphs, relationships between the concepts of maximal independent sets, minimal dominating sets and maximal irredundant sets are used to establish what is known as domination chain of parameters. 0 ir(G)  (G)  i(G)   (G)  (G)  IR(G) In this paper, starting from the concept of edge cover, six graph theoretic parameters are introduced which obey a chain of inequalities, called as the edge covering chain of the graph G

The Cover Time of a Biased Random Walk on a Random Regular Graph of Odd Degree

We consider a random walk process on graphs introduced by Orenshtein and Shinkar (2014). At any time, the random walk moves from its current position along a previously unvisited edge chosen uniformly at random, if such an edge exists. Otherwise, it walks along a previously visited edge chosen uniformly at random. For the random $r$-regular graph, with $r$ a constant odd integer, we show that this random walk process has asymptotic vertex and edge cover times $\frac{1}{r-2}n\log n$ and $\frac{r}{2(r-2)}n\log n$, respectively, generalizing a result of Cooper, Frieze and the author (2018) from $r = 3$ to any odd $r\geqslant 3$. The leading term of the asymptotic vertex cover time is now known for all fixed $r\geqslant 3$, with Berenbrink, Cooper and Friedetzky (2015) having shown that $G_r$ has vertex cover time asymptotic to $\frac{rn}{2}$ when $r\geqslant 4$ is even.