arXiv subjects

Zekhaya B. Shozi

Publications and source records attributed to Zekhaya B. Shozi.

8 recordsLinked to original sources

arXiv2026

Ratio of the number of $\mathbf{1}$-nearly independent vertex subsets and the Merrifield-Simmons index

The number $σ_k(G)$ of induced subgraphs with size $k$ of a graph $G$ was introduced recently as the number of $k$-nearly independent vertex subsets of $G$. Results highlighting similarity and difference in the behaviours of $σ_1$ and $σ_0$, have been reported. In this paper, we provide more comparison tools, by studying the ratio $\frac{σ_1(G)}{σ_{0}(G)}$. We establish sharp lower and upper bounds for this ratio over various classes of graphs, including connected graphs, trees, and forests.

math.CO↗

arXiv2026

Nordhaus-Gaddum inequalities for the number of 1-nearly independent vertex subsets

For a graph $G$, a vertex subset is called \emph{$1$-nearly independent} if the subgraph it induces contains exactly one edge. Let $σ_1(G)$ denote the number of such subsets in $G$. In this paper, we study Nordhaus-Gaddum type inequalities for $σ_1$, that is, bounds on the sum $σ_1(G)+σ_1(\overline{G})$, where $\overline{G}$ denotes the complement of $G$. We establish that, for any $n$-vertex graph $G$, we have $σ_1(G)+σ_1(\overline{G})\geq n(n-1)/2,$ with equality if and only if $G$ is either complete or edgeless. We further obtain that among all trees of order $n$, the star $K_{1,n-1}$ uniquely minimises $σ_1(T)+σ_1(\overline{T})$. Finally, we prove that for all graphs of order $n \ge 6$, \[ σ_1(G)+σ_1(\overline{G}) \le \frac{27}{64}\,2^{n} + \frac{1}{2}(n+2)(n-3), \] with equality if and only if $G$ or $\overline{G}$ is isomorphic to $3K_2 \cup \overline{K_{n-6}}$.

math.CO↗

arXiv2025

On the generator graph of a cyclic group

In this paper, we continue the study of the generator graph of a group. In 2023, Tacbobo [9] defined the generator graph of a nontrivial group to be the graph whose vertices are the elements of the group, with two vertices being adjacent if at least one of them is a generator of the group. Building on the properties established in [9], we prove that the diameter of the generator graph of a cyclic group is at most $2$. Furthermore, we present explicit formulas for some topological indices of the generator graph of a cyclic group with $n \ge 2$ elements and whose set of generators is $S$, expressed in terms of $n$ and $|S|$. Lastly, we determine the metric dimension of the generator graph of a nontrivial cyclic group as a function of its order $n$.

math.CO↗

arXiv2024

An improved lower bound on the number of $1$-nearly independent vertex subsets

Let $G=(V(G),E(G))$ be a graph with set of vertices $V(G)$ and set of edges $E(G)$. For $k\ge 0$ an integer, a subset $I_k$ of $V(G)$ is called a $k$-nearly independent vertex subset of $G$ if $I_k$ induces a subgraph of size $k$ in $G$. The number of such subsets in $G$ is denoted by $σ_k(G)$. In this paper we continue the study of $σ_1$. In particular, we prove the lower bound on $σ_1$ for a connected graph that contains a cycle and also characterise the two extremal graphs. This improves the result obtained in [E. O. D. Andriantiana and Z. B. Shozi. The number of 1-nearly independent vertex subsets. \textit{Quaestiones Mathematicae}, accepted].

math.CO↗

arXiv2024

The $1$-nearly edge independence number of a graph

Let $G = (V(G), E(G))$ be a graph. The maximum cardinality of a set $M_k \subseteq E(G)$ such that $M_k$ contains exactly $k$-pairs of adjacent edges of $G$ is called the $k$-nearly edge independence number of $G$, and is denoted by $α'_k(G)$. In this paper we study $α_1'(G)$. In particular, we prove a tight lower (resp. upper) bound on $α_1(G)$ if $G$ is a graph with given number of vertices. Furthermore, we present a characterisation of the general (resp. connected) graphs with given number of vertices and smallest $1$-nearly edge independence number. Lastly, we pose an open problem for further exploration of this study.

math.CO↗

arXiv2024

The $1$-nearly vertex independence number of a graph

Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$. A set $I_0(G) \subseteq V(G)$ is a vertex independent set if no two vertices in $I_0(G)$ are adjacent in $G$. We study $α_1(G)$, which is the maximum cardinality of a set $I_1(G) \subseteq V(G)$ that contains exactly one pair of adjacent vertices of $G$. We call $I_1(G)$ a $1$-nearly vertex independent set of $G$ and $α_1(G)$ a $1$-nearly vertex independence number of $G$. We provide some cases of explicit formulas for $α_1$. Furthermore, we prove a tight lower (resp. upper) bound on $α_1$ for graphs of order $n$. The extremal graphs that achieve equality on each bound are fully characterised.

math.CO↗

arXiv2024

The number of 1-nearly independent edge subsets

Let $G=(V(G),E(G))$ be a graph with set of vertices $V(G)$ and set of edges $E(G)$. A subset $S$ of $E(G)$ is called a $k$-nearly independent edge subsets if there are exactly $k$ pairs of elements of $S$ that share a common end. $Z_k(G)$ is the number of such subsets. This paper studies $Z_1$. Various properties of $Z_1$ are discussed. We characterise the two $n$-vertex trees with smallest $Z_1$, as well as the one with largest value. A conjecture on the $n$-vertex tree with second-largest $Z_1$ is proposed.

math.CO↗

arXiv2023

The number of $1$-nearly independent vertex subsets

Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$. A subset $I$ of $V(G)$ is an independent vertex subset if no two vertices in $I$ are adjacent in $G$. We study the number, $σ_1(G)$, of all subsets of $v(G)$ that contain exactly one pair of adjacent vertices. We call those subsets 1-nearly independent vertex subsets. Recursive formulas of $σ_1$ are provided, as well as some cases of explicit formulas. We prove a tight lower (resp. upper) bound on $σ_1$ for graphs of order $n$. We deduce as a corollary that the star $K_{1,n-1}$ (the tree with degree sequence $(n-1,1,\dots,1)$) is the $n$-vertex tree with smallest $σ_1$, while it is well known that $K_{1,n-1}$ is the $n$-vertex tree with largest number of independent subsets.

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