Median eigenvalues of bipartite graphs

For a graph $G$ of order $n$ and with eigenvalues $\lambda_1\geqslant\cdots\geqslant\lambda_n$, the HL-index $R(G)$ is defined as $R(G) ={\max}\left\{|\lambda_{\lfloor(n+1)/2\rfloor}|, |\lambda_{\lceil(n+1)/2\rceil}|\right\}.$ We show that for every connected bipartite graph $G$ with maximum degree $\Delta\geqslant3$, $R(G)\leqslant\sqrt{\Delta-2}$ unless $G$ is the the incidence graph of a projective plane of order $\Delta-1$. We also present an approach through graph covering to construct infinite families of bipartite graphs with large HL-index

Star complements in regular graphs: Old and new results

We survey results concerning star complements in finite regular graphs, and note the connection with designs and strongly regular graphs in certain cases. We include improved proofs along with new results on stars and windmills as star complements

Interval minors of complete bipartite graphs

Interval minors of bipartite graphs were recently introduced by Jacob Fox in the study of Stanley-Wilf limits. We investigate the maximum number of edges in $K_{r,s}$-interval minor free bipartite graphs. We determine exact values when $r=2$ and describe the extremal graphs. For $r=3$, lower and upper bounds are given and the structure of $K_{3,s}$-interval minor free graphs is studied

The weak saturation number of K2,t\boldsymbol{K_{2, t}}

For two graphs $G$ and $F$, we say that $G$ is weakly $F$-saturated if $G$ contains no copy of $F$ as a subgraph and one could join all the nonadjacent pairs of vertices of $G$ in some order so that a new copy of $F$ is created at each step. The weak saturation number $\mathrm{wsat}(n, F)$ is the minimum number of edges of a weakly $F$-saturated graph on $n$ vertices. In this paper, we examine $\mathrm{wsat}(n, K_{s, t})$, where $K_{s, t}$ is the complete bipartite graph with parts of sizes $s$ and $t$. We determine $\mathrm{wsat}(n, K_{2, t})$, correcting a previous report in the literature. It is also shown that $\mathrm{wsat}(s+t, K_{s,t})=\binom{s+t-1}{2}$ if $\gcd(s, t)=1$ and $\mathrm{wsat}(s+t, K_{s,t})=\binom{s+t-1}{2}+1$, otherwise

Bootstrap percolation on the Hamming graphs

The $r$-edge bootstrap percolation on a graph is an activation process of the edges. The process starts with some initially activated edges and then, in each round, any inactive edge whose one of endpoints is incident to at least $r$ active edges becomes activated. A set of initially activated edges leading to the activation of all edges is said to be a percolating set. Denote the minimum size of a percolating set in the $r$-edge bootstrap percolation process on a graph $G$ by $m_e(G, r)$. The importance of the $r$-edge bootstrap percolation relies on the fact that $m_e(G, r)$ provides bounds on $m(G, r)$, that is, the minimum size of a percolating set in the $r$-neighbor bootstrap percolation process on $G$. In this paper, we explicitly determine $m_e(K_n^d, r)$, where $K_n^d$ is the Cartesian product of $d$ copies of the complete graph on $n$ vertices which is referred as Hamming graph. Using this, we show that $m(K_n^d, r)=(1+o(1))\frac{d^{r-1}}{r!}$ when $n, r$ are fixed and $d$ goes to infinity which extends a known result on hypercubes

Weak saturation numbers in random graphs

For two given graphs $G$ and $F$, a graph $H$ is said to be weakly $(G, F)$-saturated if $H$ is a spanning subgraph of $G$ which has no copy of $F$ as a subgraph and one can add all edges in $E(G)\setminus E(H)$ to $H$ in some order so that a new copy of $F$ is created at each step. The weak saturation number $wsat(G, F)$ is the minimum number of edges of a weakly $(G, F)$-saturated graph. In this paper, we deal with the relation between $wsat(G(n,p), F)$ and $wsat(K_n, F)$, where $G(n,p)$ denotes the Erd\H{o}s--R\'enyi random graph and $K_n$ denotes the complete graph on $n$ vertices. For every graph $F$ and constant $p$, we prove that $wsat( G(n,p),F)= wsat(K_n,F)(1+o(1))$ with high probability. Also, for some graphs $F$ including complete graphs, complete bipartite graphs, and connected graphs with minimum degree $1$ or $2$, it is shown that there exists an $\varepsilon(F)>0$ such that, for any $p\geqslant n^{-\varepsilon(F)}\log n$, $wsat( G(n,p),F)= wsat(K_n,F)$ with high probability

Hadamard Matrices with Few Distinct Types

The notion of type of quadruples of rows is proven to be useful in the classification of Hadamard matrices. We investigate Hadamard matrices with few distinct types. Apparently, Hadamard matrices with few distinct types are very rare and have nice combinatorial properties. We show that there exists no Hadamard matrix of order larger than $12$ whose quadruples of rows are all of the same type. We then focus on Hadamard metrics with two distinct types. Among other results, the Sylvester Hadamard matrices are shown to be characterized by their spectrum of types. This is a joint work with A. Mohammadian.Non UBCUnreviewedAuthor affiliation: Institute for Research in Fundamental Sciences (IPM)Facult

On the unimodality of independence polynomial of certain classes of graphs

The independence polynomial of a graph G is the polynomial $sum i_kx^k$, where $i_k$ denote the number of independent sets of cardinality k in G. In this paper we study unimodality problem for the independence polynomial of certain classes of graphs