arXiv subjects

Ali Mohammadian

Publications and source records attributed to Ali Mohammadian.

9 recordsLinked to original sources

arXiv2026

$\boldsymbol{2}$-Neighbor Bootstrap Percolation on Odd Graphs

The $r$-neighbor bootstrap percolation process on a graph $G$ is a vertex-activation process that begins with a set of initially active vertices. In each subsequent round, every inactive vertex having at least $r$ active neighbors becomes active. Denote by $m(G,r)$ the minimum number of initially active vertices whose activation eventually spreads to all vertices of $G$. Let $\mathbbmsl{O}_k$ and $\mathbbmsl{B}_k$ be the odd graph and the bipartite odd graph, respectively, both defined on a ground set of size $2k+1$. In this article, among other results, we prove that $m(\mathbbmsl{O}_k, 2)=\lceil(k^2+3k+2)/4\rceil$ and $m(\mathbbmsl{B}_k, 2)=\lceil(k^2+2k+3)/2\rceil$. The former result confirms a conjecture posed in 2021 by Grippo, Pastine, Torres, Valencia-Pabon, and Vera.

math.CO↗

arXiv2026

A search for Hadamard matrices of Williamson type

In this article, we consider a special class of Williamson type matrices which we call them near Williamson matrices. They are in fact four $n\times n$ $(-1, 1)$-matrices $A, B, C, D$ so that $A$ is circulant, $B,C,D$ are symmetric circulant, and they satisfy $AA^\top+BB^\top+CC^\top+DD^\top=4nI$. Using a computer search, we find all inequivalent near Williamson matrices for all odd orders at most $35$. We also show that such matrices exist for all odd orders up to $63$. As a consequence, we find the first known example of a quaternary Hadamard matrix of order $118$.

math.CO↗

arXiv2025

On saturation numbers of complete multipartite graphs and even cycles

Given positive integer $n$ and graph $F$, the saturation number $\mathrm{sat}(n, F)$ is the minimum number of edges in an edge-maximal $F$-free graph on $n$ vertices. In this paper, we determine asymptotic behavior of $\mathrm{sat}(n, F)$ when $F$ is either a complete multipartite graph or a cycle graph whose length is even and large enough. This extends a result by Bohman, Fonoberova, and Pikhurko from 2010 as well as partially resolves a conjecture of Füredi and Kim from 2013.

math.CO↗

arXiv2024

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 Erdős--Rényi 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.

math.CO↗

arXiv2024

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.

math.CO↗

arXiv2024

On the classification of skew Hadamard matrices of order 36 and related structures

Two skew Hadamard matrices are considered {\sf SH}-equivalent if they are similar by a signed permutation matrix. This paper determines the number of {\sf SH}-inequivalent skew Hadamard matrices of order $36$ for some types. We also study ternary self-dual codes and association schemes constructed from the skew Hadamard matrices of order $36$.

math.CO↗

arXiv2024

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

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.

math.CO↗

arXiv2023

Saturation numbers of bipartite graphs in random graphs

For a given graph $F$, the $F$-saturation number of a graph $G$, denoted by $ {sat}(G, F)$, is the minimum number of edges in an edge-maximal $F$-free subgraph of $G$. In 2017, Korándi and Sudakov determined $ {sat}({G}(n, p), K_r)$ asymptotically, where ${G}(n, p) $ denotes the Erdős-Rényi random graph and $ K_r$ is the complete graph on $r$ vertices. In this paper, among other results, we present an asymptotic upper bound on ${sat}({G}(n, p), F)$ for any bipartite graph $F$ and also an asymptotic lower bound on ${sat}({G}(n, p), F)$ for any complete bipartite graph $F$.

math.CO↗

arXiv2021

On the location of zeros of the Laplacian matching polynomials of graphs

The Laplacian matching polynomial of a graph $G$, denoted by $\mathscr{L\hspace{-0.7mm}M}(G,x)$, is a new graph polynomial whose all roots are nonnegative real numbers. In this paper, we investigate the location of zeros of the Laplacian matching polynomials. Let $G$ be a connected graph. We show that $0$ is a root of $\mathscr{L\hspace{-0.7mm}M}(G, x)$ if and only if $G$ is a tree. We prove that the number of distinct positive zeros of $\mathscr{L\hspace{-0.7mm}M}(G,x)$ is at least equal to the length of the longest path in $G$. It is also established that the zeros of $\mathscr{L\hspace{-0.7mm}M}(G,x)$ and $\mathscr{L\hspace{-0.7mm}M}(G-e,x)$ interlace for each edge $e$ of $G$. Using the path-tree of $G$, we present a linear algebraic approach to investigate the largest zero of $\mathscr{L\hspace{-0.7mm}M}(G,x)$ and particularly to give tight upper and lower bounds on it.

math.CO↗