arXiv subjects

Saeid Alikhani

Publications and source records attributed to Saeid Alikhani.

At least 37 records · Page 2Linked to original sources

arXiv2024

$k$-Coalitions in Graphs

In this paper, we propose and investigate the concept of $k$-coalitions in graphs, where $k\ge 1$ is an integer. A $k$-coalition refers to a pair of disjoint vertex sets that jointly constitute a $k$-dominating set of the graph, meaning that every vertex not in the set has at least $k$ neighbors in the set. We define a $k$-coalition partition of a graph as a vertex partition in which each set is either a $k$-dominating set with exactly $k$ members or forms a $k$-coalition with another set in the partition. The maximum number of sets in a $k$-coalition partition is called the $k$-coalition number of the graph represented by $C_k(G)$. We present fundamental findings regarding the properties of $k$-coalitions and their connections with other graph parameters. We obtain the exact values of $2$-coalition number of some specific graphs and also study graphs with large $2$-coalition number.

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arXiv2024

Golden ratio in graph theory: A survey

Much has been written about the golden ratio $ϕ=\frac{1+\sqrt{5}}{2}$ and this strange number appears mysteriously in many mathematical calculations. In this article, we review the appearance of this number in the graph theory. More precisely, we review the relevance of this number in topics such as the number of spanning trees, topological indices, energy, chromatic roots, domination roots and the number of domatic partitions of graphs.

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arXiv2024

Counting the Number of Domatic Partition of a Graph

A subset of vertices $S$ of a graph $G$ is a dominating set if every vertex in $V \setminus S$ has at least one neighbor in $S$. A domatic partition is a partition of the vertices of a graph $G$ into disjoint dominating sets. The domatic number $d(G)$ is the maximum size of a domatic partition. Suppose that $dp(G,i)$ is the number of distinct domatic partition of $G$ with cardinality $i$. In this paper, we consider the generating function of $dp(G,i)$, i.e., $DP(G,x)=\sum_{i=1}^{d(G)}dp(G,i)x^i$ which we call it the domatic partition polynomial. We explore the domatic polynomial for trees, providing a quadratic time algorithm for its computation based on weak 2-coloring numbers. Our results include specific findings for paths and certain graph products, demonstrating practical applications of our theoretical framework.

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arXiv2024

Elliptic Sombor energy of a graph

Let $G$ be a simple graph with vertex set $V(G) = \{v_1, v_2,\ldots, v_n\}$. The elliptic Sombor matrix of $G$, denoted by $A_{ESO}(G)$, is defined as the $n\times n$ matrix whose $(i,j)$-entry is $(d_i+d_j)\sqrt{d_i^2+d_j^2}$ if $v_i$ and $v_j$ are adjacent and $0$ for another cases. Let the eigenvalues of the elliptic Sombor matrix $A_{ESO}(G)$ be $ρ_1\geq ρ_2\geq \ldots\geq ρ_n$ which are the roots of the elliptic Sombor characteristic polynomial $\prod_{i=1}^n (ρ-ρ_i)$. The elliptic Sombor energy ${E_{ESO}}$ of $G$ is the sum of absolute values of the eigenvalues of $A_{ESO}(G)$. In this paper, we compute the elliptic Sombor characteristic polynomial and the elliptic Sombor energy for some graph classes. We compute the elliptic Sombor energy of cubic graphs of order $10$ and as a consequence, we see that two $k$-regular graphs of the same order may have different elliptic Sombor energy.

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arXiv2024

Domination polynomial and total domination polynomial of zero-divisor graphs of commutative rings

The domination polynomial (the total domination polynomial) of a graph $ G $ of order $ n $ is the generating function of the number of dominating sets (total dominating sets) of $ G $ of any size. In this paper, we study the domination polynomial and the total domination polynomial of zero-divisor graphs of the ring $ \mathbb{Z}_n $ where $ n\in\lbrace 2p, p^2, pq, p^2q, pqr, p^α\rbrace $, and $ p, q, r $ are primes with $ p>q>r>2 $.

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arXiv2024

Pebbling number of polymers

Let $G=(V,E)$ be a simple graph. A function $f:V\rightarrow \mathbb{N}\cup \{0\}$ is called a configuration of pebbles on the vertices of $G$ and the quantity $\vert f\vert=\sum_{u\in V}f(u)$ is called the weight of $f$ which is just the total number of pebbles assigned to vertices. A pebbling step from a vertex $u$ to one of its neighbors $v$ reduces $f(u)$ by two and increases $f(v)$ by one. A pebbling configuration $f$ is said to be solvable if for every vertex $ v $, there exists a sequence (possibly empty) of pebbling moves that results in a pebble on $v$. The pebbling number $ π(G) $ equals the minimum number $ k $ such that every pebbling configuration $ f $ with $ \vert f\vert = k $ is solvable. Let $ G $ be a connected graph constructed from pairwise disjoint connected graphs $ G_1,...,G_k $ by selecting a vertex of $ G_1 $, a vertex of $ G_2 $, and identifying these two vertices. Then continue in this manner inductively. We say that $ G $ is a polymer graph, obtained by point-attaching from monomer units $ G_1,...,G_k $. In this paper, we study the pebbling number of some polymers.

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arXiv2024

On the number of outer connected dominating sets of graphs

Let $G=(V,E)$ be a simple graph. A set $S\subseteq V(G)$ is called an outer-connected dominating set (or ocd-set) of $G$, if $S$ is a dominating set of $G$ and either $S=V(G)$ or $V\backslash S$ is a connected graph. In this paper we introduce a polynomial which its coefficients are the number of ocd-sets of $G$. We obtain some properties of this polynomial and its coefficients. Also we compute this polynomial for some specific graphs.

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arXiv2024

More on graph pebbling number

Let $G=(V,E)$ be a simple graph. A function $ϕ:V\rightarrow \mathbb{N}\cup \{0\}$ is called a configuration of pebbles on the vertices of $G$ and the quantity $\sum_{u\in V}ϕ(u)$ is called the size of $ϕ$ which is just the total number of pebbles assigned to vertices. A pebbling step from a vertex $u$ to one of its neighbors $v$ reduces $ϕ(u)$ by two and increases $ϕ(v)$ by one. Given a specified target vertex $r$ we say that $ϕ$ is $t$-fold $r$-solvable, if some sequence of pebbling steps places at least $t$ pebbles on $r$. Conversely, if no such steps exist, then $ϕ$ is $r$-unsolvable. The minimum positive integer $m$ such that every configuration of size $m$ on the vertices of $G$ is $t$-fold $r$-solvable is denoted by $π_t(G,r)$. The $t$-fold pebbling number of $G$ is defined to be $π_t(G)= max_{r\in V(G)}π_t(G,r)$. When $t=1$, we simply write $π(G)$, which is the pebbling number of $G$. In this note, we study the pebbling number for some specific graphs. Also we investigate the pebbling number of corona and neighbourhood corona of two graphs.

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arXiv2024

A New Algorithm for Computing the Frobenius Number

A number $α$ has a representation with respect to the numbers $α_1,...,α_n$, if there exist the non-negative integers $λ_1,... ,λ_n$ such that $α=λ_1α_1+...+λ_n α_n$. The largest natural number that does not have a representation with respect to the numbers $α_1,...,α_n$ is called the Frobenius number and is denoted by the symbol $g(α_1,...,α_n)$. In this paper, we present a new algorithm to calculate the Frobenius number. Also we present the sequential form of the new algorithm.

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arXiv2024

The number of distinguishing colorings of a Cartesian product graph

A vertex coloring is called distinguishing if the identity is the only automorphism that can preserve it. The distinguishing threshold $θ(G)$ of a graph $G$ is the minimum number of colors $k$ required that any arbitrary $k$-coloring of $G$ is distinguishing. In this paper, we calculate the distinguishing threshold of a Cartesian product graph. Moreover, we calculate the number of non-equivalent distinguishing colorings of grids.

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arXiv2024

Co-even Domination Number of a Modified Graph by Operations on a Vertex or an Edge

Let $G=(V,E)$ be a simple graph. A dominating set of $G$ is a subset $D\subseteq V$ such that every vertex not in $D$ is adjacent to at least one vertex in $D$. The cardinality of a smallest dominating set of $G$, denoted by $γ(G)$, is the domination number of $G$. A dominating set $D$ is called co-even dominating set if the degree of vertex $v$ is even number for all $v\in V-D$. The cardinality of a smallest co-even dominating set of $G$, denoted by $γ_{coe}(G)$, is the co-even domination number of $G$. In this paper we study co-even domination number of graphs which constructed by some operations on a vertex or an edge of a graph.

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arXiv2023

Independent domination stability in graphs

A non-empty set $S\subseteq V (G)$ of the simple graph $G=(V(G),E(G))$ is an independent dominating set of $G$ if every vertex not in $S$ is adjacent with some vertex in $S$ and the vertices of $S$ are pairwise non-adjacent. The independent domination number of $G$, denoted by $γ_i(G)$, is the minimum size of all independent dominating sets of $G$. The independent domination stability, or simply $id$-stability of $G$ is the minimum number of vertices whose removal changes the independent domination number of $G$. In this paper, we investigate properties of independent domination stability in graphs. In particular, we obtain several bounds and obtain the independent domination stability of some operations of two graphs.

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arXiv2023

More on the $2$-restricted optimal pebbling number

Let $G=(V,E)$ be a simple graph. A function $f:V\rightarrow \mathbb{N}\cup \{0\}$ is called a configuration of pebbles on the vertices of $G$ and the weight of $f$ is $w(f)=\sum_{u\in V}f(u)$ which is just the total number of pebbles assigned to vertices. A pebbling step from a vertex $u$ to one of its neighbors $v$ reduces $f(u)$ by two and increases $f(v)$ by one. A pebbling configuration $f$ is said to be solvable if for every vertex $ v $, there exists a sequence (possibly empty) of pebbling moves that results in a pebble on $v$. A pebbling configuration $f$ is a $t$-restricted pebbling configuration (abbreviated $t$RPC) if $f(v)\leq t$ for all $v\in V$. The $t$-restricted optimal pebbling number $π_t^*(G)$ is the minimum weight of a solvable $t$RPC on $G$. Chellali et.al. [Discrete Appl. Math. 221 (2017) 46-53] characterized connected graphs $G$ having small $2$-restricted optimal pebbling numbers and characterization of graphs $G$ with $π_2^*(G)=5$ stated as an open problem. In this paper, we solve this problem. We improve the upper bound of the $2$-restricted optimal pebbling number of trees of order $n$. Also, we study $2$-restricted optimal pebbling number of some grid graphs, corona and neighborhood corona of two specific graphs.

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arXiv2023

Strong domination number of graphs from primary subgraphs

A set $D$ of vertices is a strong dominating set in a graph $G$, if for every vertex $x\in V(G) \setminus D$ there is a vertex $y\in D$ with $xy\in E(G)$ and $deg(x) \leq deg(y)$. The strong domination number $γ_{st}(G)$ of $G$ is the minimum cardinality of a strong dominating set in $G$. Let $G$ be a connected graph constructed from pairwise disjoint connected graphs $G_1,\ldots ,G_k$ by selecting a vertex of $G_1$, a vertex of $G_2$, and identifying these two vertices, and thereafter continuing in this manner inductively. The graphs $G_1,\ldots ,G_k$ are the primary subgraphs of $G$. In this paper, we study the strong domination number of $K_r$-gluing of two graphs and investigate the strong domination number for some particular cases of graphs from their primary subgraphs.

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arXiv2023

Strong domatic number of a graph

A set $D$ of vertices of a simple graph $G=(V,E)$ is a strong dominating set, if for every vertex $x\in \overline{D}=V\setminus D$ there is a vertex $y\in D$ with $xy\in E(G)$ and $deg(x)\leq deg(y)$. The strong domination number $γ_{st}(G)$ is defined as the minimum cardinality of a strong dominating set. The strong domatic number of $G$ is the maximum number of strong dominating sets into which the vertex set of $G$ can be partitioned. We initiate the study of the strong domatic number, and we present different sharp bounds on $d_{st}(G)$. In addition, we determine this parameter for some classes of graphs, such as cubic graphs of order at most $10$.

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arXiv2023

Connected coalitions in graphs

The connected coalition in a graph $G=(V,E)$ consists of two disjoint sets of vertices $V_{1}$ and $V_{2}$, neither of which is a connected dominating set but whose union $V_{1}\cup V_{2}$, is a connected dominating set. A connected coalition partition in a graph $G$ of order $n=|V|$ is a vertex partition $ψ$ = $\{V_1, V_2,..., V_k \}$ such that every set $V_i \in ψ$ either is a connected dominating set consisting of a single vertex of degree $n-1$, or is not a connected dominating set but forms a connected coalition with another set $V_j\in ψ$ which is not a connected dominating set. The connected coalition number, denoted by $CC(G)$, is the maximum cardinality of a connected coalition partition of $G$. In this paper, we initiate the study of connected coalition in graphs and present some basic results. Precisely, we characterize all graphs that have a connected coalition partition. Moreover, we show that for any graph $G$ of order $n$ with $δ(G)=1$ and with no full vertex, it holds that $CC(G)<n$. Furthermore, we show that for any tree $T$, $CC(T)=2$. Finally, we present two polynomial-time algorithms that for a given connected graph $G$ of order $n$ determine whether $CC(G)=n$ or $CC(G)=n-1$.

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arXiv2023

Strong domination number of Hajós sum and vertex-sum of two graphs

Let $G=(V,E)$ be a simple graph. A set $D\subseteq V$ is a strong dominating set of $G$, if for every vertex $x\in V\setminus D$ there is a vertex $y\in D$ with $xy\in E(G)$ and $deg(x)\leq deg(y)$. The strong domination number $γ_{st}(G)$ is defined as the minimum cardinality of a strong dominating set. In this paper, we study the strong domination number of Hajós sum and vertex-sum of two graphs.

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arXiv2022

Coalition of cubic graphs of order at most $10$

The coalition in a graph $G$ consists of two disjoint sets of vertices $V_{1}$ and $V_{2}$, neither of which is a dominating set but whose union $V_{1}\cup V_{2}$, is a dominating set. A coalition partition in a graph $G$ is a vertex partition $π$ = $\{V_1, V_2,..., V_k \}$ such that every set $V_i \in π$ is not a dominating set but forms a coalition with another set $V_j\in π$ which is not a dominating set. The coalition number $C(G)$ equals the maximum $k$ of a coalition partition of $G$. In this paper, we compute the coalition number of all cubic graphs of order at most $10$.

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