The competition graph of a doubly partial order is known to be an interval graph. The CCE graph and the niche graph of a doubly partial order are also known to be interval graphs if the graphs do not contain a cycle of length four and three as an induced subgraph, respectively. Phylogeny graphs are variant of competition graphs. The phylogeny graph P(D) of a digraph D is the (simple undirected) graph defined by V (P(D)) := V (D) and E(P(D)) := {xy | N+D (x) ∩ N+D(y) ¹ ⊘ } ⋃ {xy | (x,y) ∈ A(D)},...

A nonincreasing sequence $\pi =(d_1,...,d_n)$ of nonnegative integers is a graphic sequence if it is realizable by a simple graph $G$ on $n$ vertices. In this case, $G$ is referred to as a realization of $\pi$. Given two graphs $G_1$ and $G_2$, A. Busch et al. (2014) introduced the potential-Ramsey number of $G_1$ and $G_2$, denoted by $r_{\mathrm{pot}}(G_1,G_2)$, as the smallest nonnegative integer $m$ such that for every $m$-term graphic sequence $\pi$, there is a realization $G$ of $\pi$ with $G_1\subseteq G$ or with $G_2\subseteq \overline{G}$, where $\overline{G}$ is the complement of $G$. For $t\ge 2$ and $0\le k\le \lfloor \frac{t}{2}\rfloor$, let $K_t^{-k}$ be the graph obtained...

Let $R$ be a commutative ring with unity and $U\left(R\right)$ be the set of unit elements of $R$. In this paper, we introduce and investigate some properties of a new kind of graph on the ring $R$, namely, the prime ideals intersection graph of $R$, denoted by $G_p\left(R\right)$. The $G_p\left(R\right)$ is a graph with vertex set $R^{*}-U\left(R\right)$ and two distinct vertices $a$ and $b$ are adjacent if and only if there exists a prime ideal $𝔭$ of $R$ such that $a,b\in 𝔭$. We obtain necessary and sufficient conditions on $R$ such that $G_p\left(R\right)$ is disconnected. We find the diameter and girth of $G_p\left(R\right)$....

The scrambling index of an $n\times n$ primitive Boolean matrix $A$ is the smallest positive integer $k$ such that $A^k{\left(A^{\mathrm{T}}\right)}^k=J$, where $A^{\mathrm{T}}$ denotes the transpose of $A$ and $J$ denotes the $n\times n$ all ones matrix. For an $m\times n$ Boolean matrix $M$, its Boolean rank $b\left(M\right)$ is the smallest positive integer $b$ such that $M=AB$ for some $m\times b$ Boolean matrix $A$ and $b\times n$ Boolean matrix $B$. In 2009, M. Akelbek, S. Fital, and J. Shen gave an upper bound on the scrambling index of an $n\times n$ primitive matrix $M$ in terms of its Boolean rank $b\left(M\right)$, and they also characterized all primitive...