arXiv subjects

Saeid Alikhani

Publications and source records attributed to Saeid Alikhani.

At least 19 recordsLinked to original sources

arXiv2026

On the Integer Domination Root Conjecture

The domination integer root conjecture asserted that $0$ and $-2$ are the only integer roots of the domination polynomial $D(G, x)$ for any graph $G$. In this paper, we document a counterexample of order $n = 33$ possessing an integer domination root at $x = -4$. We provide the complete structural description of the graph $G_{33}$, present its exact domination polynomial $D(G_{33}, x)$, and demonstrate its exact rational factorization. Furthermore, we outline the structural gadget mechanism involving transfer matrices and $S$-unit branch cancellations that gives rise to non-trivial zero evaluation at $x = -4$.

math.CO↗

arXiv2026

Super Coalition Number in Graphs

We introduce and investigate the structural properties of super coalition partitions in graphs, a novel direction that bridges cooperative resource deployment with rigid domination criteria. Based on the foundational concept of super domination, a super coalition partition is defined as a vertex set partitioning $\Upsilon = \{A_1, A_2, \ldots, A_k\}$ such that no single class $A_i$ constitutes a valid super dominating set, yet every class can be paired with at least one distinct partner class $A_j$ to form a union $A_i \cup A_j$ that achieves full super domination over the graph. The super coalition number, denoted by $C_s(G)$, represents the maximum possible cardinality of such a partition. In this paper, we establish general operational bounds for $C_s(G)$ using the underlying order and the super domination number $\gamma_{sp}(G)$, demonstrate its relation to the super domatic number $d_{sp}(G)$, analyze its computational complexity proving its NP-complete nature under general conditions, and provide exact determinations for key standard graph architectures including paths, cycles, complete graphs, stars, wheels, and friendship configurations. We conclude by proving that the super coalition number can grow arbitrarily large.

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arXiv2026

On the Spectrum of the Line Graph of a Family of Bipartite Graphs Arising from the Boolean Lattice

The Boolean lattice $BL_n$, $n\geq 3$, is the graph whose vertex set is the collection of all subsets of $[n]=\{1,2,\ldots,n\}$, where two subsets $U$ and $W$ are adjacent if and only if their symmetric difference has precisely one element. In the graph $BL_n$, the \emph{layer} $L_k$ is the family of all $k$-element subsets of $[n]$. The subgraph $BL_n(k-1,k)$ is the induced subgraph of $BL_n$ on layers $L_{k-1}$ and $L_{k}$. This graph is bipartite and, when $n=2k-1$, is $k$-regular and isomorphic to the bipartite double cover $2{\cdot}O_k$ of the odd graph $O_k$. In this paper, we determine the full adjacency spectrum -- eigenvalues together with their multiplicities -- of the line graph $L(BL_n(k-1,k))$ for all admissible values of $n$ and $k$. As a consequence, we show that $L(BL_n(k-1,k))$ is an integral graph whenever $n = 2k-1$, and we recover as a special case the spectrum of the line graph $L(n)$ of $BL_n(1,2)$ established by Mirafzal~\cite{pap-sm-1}.

math.CO↗

arXiv2026

A Minimum Doubly Resolving Set and Strong Resolving Set for the Crystal Cubic Carbon

The task of identifying resolving sets has been extensively studied due to its wide relevance in fields such as chemistry, robot navigation, combinatorial optimization, pattern recognition, and image processing. These applications have helped motivate and establish the theoretical foundations of the subject. Notably, problems of this type are generally known to be NP-hard. This study introduces an alternative structural representation for the crystal cubic carbon $ CCC(n) $. Building on this model, we determine the minimum sizes of both a doubly resolving set and a strong resolving set for $CCC(n)$.

math.GM↗

arXiv2026

The partition dimension and $k$-domination number of a family of non-distance regular graph

A partition $Σ= \{S_1, S_2, \dots, S_k\}$ of the vertex set $V(G)$ is a resolving partition if every pair of distinct vertices in $G$ has a unique representation relative to $Σ$. The partition dimension, $pd(G)$, is the minimum cardinality of such a partition. Additionally, a subset $D \subseteq V(G)$ is a $k$-dominating set if every vertex in $V(G) \setminus D$ has at least $k$ neighbors in $D$; the $k$-domination number, $γ_k(G)$, denotes the minimum size of such a set. Determining these parameters is NP-complete and particularly challenging for non-distance-regular graphs. This paper consider the Toeplitz graph $T_{2n}(W)$, a family of non-distance-regular graphs. While some resolving parameters for this family have been established, its partition dimension and $k$-domination number remain unknown. We close this gap by computing both parameters for $T_{2n}(W)$.

math.CO↗

arXiv2026

On the Number of Connected Edge Cover Sets of Some Graph Families

Let $G=(V,E)$ be a simple connected graph. A connected edge cover of $G$ is a subset $S\subseteq E$ such that every vertex of $G$ is incident with at least one edge in $S$ and the subgraph induced by $S$ is connected. The connected edge cover polynomial of $G$ is defined as $E_c(G,x)=\sum_{i} e_c(G,i)x^i$, where $e_c(G,i)$ denotes the number of connected edge covers of $G$ with exactly $i$ edges. In this paper, we derive explicit formulas for both the connected edge cover polynomials and the total number of connected edge covers for several important graph families, including wheels, complete graphs $K_n$, complete bipartite graphs $K_{2,n}$, friendship graphs, and lollipop graphs. Each formula is accompanied by a combinatorial proof and verified by computational enumeration for small orders.

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arXiv2026

Stability of the Strong Domination Number of Graphs

This paper introduces and studies the stability of the strong domination number of a graph, denoted $\operatorname{st}_{\gamma_{st}}(G)$, defined as the minimum number of vertices whose removal changes the strong domination number $\gamma_{st}(G)$. We determine exact values of this stability parameter for several fundamental graph classes, including paths, cycles, wheels, complete bipartite graphs, friendship graphs, book graphs, and balanced complete multipartite graphs. General bounds on $\operatorname{st}_{\gamma_{st}}(G)$ are established, along with a Nordhaus Gaddum type inequality. The behavior of stability under graph operations such as join, corona, and Cartesian product is also investigated. Structural characterizations of graphs with given stability values are provided, and several open problems and directions for future research are outlined.

math.CO↗

arXiv2025

Amicable numbers and their connection to the Euler totient function

A pair of numbers is amicable if each number equals the sum of the proper divisors of the other. This paper after exploring the history and evolution of amicable numbers, introduces a novel characterization of amicable pairs whose greatest common divisor is a power of two, using their distinct prime factorizations. Specifically, we examine pairs of the forms $A=2^n ab, B=2^n cd$, $A=2^n abc, B=2^n de$, and $A=2^n abc, B=2^n def$. From these configurations, we establish explicit symmetric identities that relate the sum $φ(A)+φ(B)$ of Euler's totient functions directly to the odd prime factors of $A$ and $B$.

math.HO↗

arXiv2025

Some results on Hamming graphs and an extended Hamming graphs

In this paper we first obtain the spectrum of the folded hypercube in a new approach. Then we introduce a new family of graphs called the extended Hamming graph, denoted by $EH(n,2^n)$, which is constructed from the well-known Hamming graph $H(n,2^n)$. The graph $EH(n,2^n)$ shares the same vertex set as $H(n,2^n)$ but includes additional edges, called complementary edges, connecting each $n$-tuple vertex $u$ to its complement $u^c$, where $u^c$ is defined such that the sum of each two corresponding coordinates of $u$ and $u^c$ equals $2^n-1$. We investigate several algebraic and structural properties of this new family of graphs. Specifically, we show that the diameter of $EH(n,2^n)$ is $n$. We prove that $EH(n,2^n)$ is a Cayley graph, but we demonstrate that it is not a distance regular graph. Finally, we determine the spectrum of $EH(n,2^n)$, showing that its eigenvalues are $λ_i\pm 1$, where $λ_i$ are the eigenvalues of the underlying Hamming graph $H(n,2^n)$. The multiplicity of each eigenvalue is explicitly calculated.

math.CO↗

arXiv2025

On the spectrum of two families of non-distance-regular graphs

This paper addresses the challenge of spectral analysis and structural investigation for graphs that are not distance-regular, where computing the spectrum using standard methods based on equitable and orbit partitions can be complex. Our main objective is to determine all eigenvalues of the extended graph $E(2.O_k)$ by leveraging the relationship between its equitable and orbit partitions. While the integral nature of this graph has been previously studied, we introduce a novel approach to demonstrate the utility of this method in finding the complete set of distinct eigenvalues for a class of non-distance-regular graphs. Specifically, we first establish that $E(2.O_k)$ is a vertex-transitive graph with diameter $k$, contrasting with the diameter of $2.O_k$, which is $2k-1$. We also determine the automorphism group of $E(2.O_k)$ and prove that it is an integral graph, meaning all eigenvalues of its adjacency matrix are integers. A significant result is the determination of the multiplicity for all distinct eigenvalues of $E(2.O_k)$. Additionally, we extend our method to the enhanced Johnson graph $EJ(2m,m)$. Although its eigenvalues are known from prior work, the multiplicity of these distinct eigenvalues has not yet been calculated. We use our techniques to fully determine the multiplicity of all distinct eigenvalues for $EJ(2m,m)$.

math.CO↗

arXiv2025

On the Harmonic characteristic polynomial of specific graphs

This paper explores the Harmonic matrix $MH(G)$ associated with a simple graph $ G $, where each entry corresponds to $ \frac{2}{d_i + d_j} $ for adjacent vertices $ v_i $ and $ v_j $. We investigate the spectral properties of this matrix, particularly focusing on its eigenvalues. A central objective of this work is to compute the Harmonic characteristic polynomial. Furthermore, we analyze the Harmonic energy $ HE(G) $ of a graph as the sum of the absolute values of the eigenvalues of $ MH(G) $. Explicit expressions for both the Harmonic characteristic polynomial and the Harmonic energy are derived for several specific classes of graphs.

math.CO↗

arXiv2025

Fair coalition in graphs

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 $G$. For $k \geq 1$, a $k$-fair dominating set ($kFD$-set) in $G$, is a dominating set $S$ such that $|N(v) \cap D|=k$ for every vertex $ v \in V\setminus D$. A fair dominating set in $G$ is a $kFD$-set for some integer $k\geq 1$. We consider $1FD$-sets and define a fair coalition in a graph $G$ as a pair of disjoint subsets $A_1, A_2 \subseteq A$ that satisfy the following conditions: (a) neither $A_1$ nor $A_2$ constitutes a $1$-fair dominating set of $G$, and (b) $A_1\cup A_2$ constitutes a $1$-fair dominating set of $G$. A fair coalition partition of a graph $G$ is a partition $Υ= \{A_1,A_2,\ldots,A_k\}$ of its vertex set such that every set $A_i$ of $Υ$ is either a singleton $1$-fair dominating set of $G$, or is not a $1$-fair dominating set of $G$ but forms a fair coalition with another non-$1$-fair dominating set $A_j\in Υ$. We define the fair coalition number of $G$ as the maximum cardinality of a fair coalition partition of $G$, and we denote it by $\mathcal{C}_f(G)$. We initiate the study of the fair coalition in graphs and obtain $\mathcal{C}_f(G)$ for some specific graphs.

math.CO↗

arXiv2025

$k$-Fair Coalitions in Graphs

Let $G = (V,E)$ be a simple graph. A subset $S \subseteq V$ is called a $k$-fair dominating set if every vertex not in $S$ has exactly $k$ neighbors in $S$. Two disjoint sets $A, B \subseteq V$ form a $k$-fair coalition of $G$ if neither $A$ nor $B$ is a $k$-fair dominating set and the union $A \cup B$ is a $k$-fair dominating set of $G$. A partition $π= \{V_1, V_2, \ldots, V_m\}$ of $V$ is called a $k$-fair coalition partition, if every set $V_i\inπ$, either $V_i$ is a $k$-fair dominating set with exactly $k$ vertices, or $V_i$ is not a $k$-fair dominating set, but forms a $k$-fair coalition with some other set $V_j$ in $π$. The $k$-fair coalition number $C_{kf}(G)$ is the largest possible size of a $k$-fair coalition partition for $G$. The objective of this study is to initiate an examination into the notion of $k$-fair coalitions in graphs and present essential findings.

math.CO↗

arXiv2025

Stability of $2$-domination number of a graph

This paper delves into the stability of the $2$-domination number in simple undirected graphs. The $2$-domination number of a graph $G$, $γ_2(G)$, represents the minimum size of a vertex subset where every other vertex in the graph is adjacent to at least two members of the subset. We define the $2$-domination stability, $st_{γ_2}(G)$, as the smallest number of vertices whose removal causes a change in $γ_2(G)$. Our primary contributions include computing this parameter for specific graphs, establishing various bounds for this stability and determining its behavior under certain graph operations combining two graphs.

math.CO↗

arXiv2025

$2$-Restricted Optimal Pebbling Number of Some Graphs

Let $G=(V,E)$ be a simple graph. A pebbling configuration on $G$ is a function $f:V\rightarrow \mathbb{N}\cup \{0\}$ that assigns a non-negative integer number of pebbles to each vertex. The weight of a configuration $f$ is $w(f)=\sum_{u\in V}f(u)$, the total number of pebbles. A pebbling move consists of removing two pebbles from a vertex $u$ and placing one pebble on an adjacent vertex $v$. A configuration $f$ is a $t$-restricted pebbling configuration ($t$RPC) if no vertex has more than $t$ pebbles. The $t$-restricted optimal pebbling number $π_t^*(G)$ is the minimum weight of a $t$RPC on $G$ that allows any vertex to be reached by a sequence of pebbling moves. The distinguishing number $D(G)$ is the minimum number of colors needed to label the vertices of $G$ such that the only automorphism preserving the coloring is the trivial one (i.e., the identity map). In this paper, we investigate the $2$-restricted optimal pebbling number of trees $T$ with $D(T)=2$ and radius at most $2$ and enumerate their $2$-restricted optimal pebbling configurations. Also we study the $2$-restricted optimal pebbling number of some graphs that are of importance in chemistry such as some alkanes.

math.CO↗

arXiv2024

On the number of connected edge cover sets in a graph

Let $ G=(V,E) $ be a simple graph of order $ n $ and size $ m $. A connected edge cover set of a graph is a subset $S$ of edges such that every vertex of the graph is incident to at least one edge of $S$ and the subgraph induced by $S$ is connected. We initiate the study of the number of the connected edge cover sets of a graph $G$ with cardinality $i$, $ e_{c}(G,i) $ and consider the generating function for $ e_{c}(G,i) $ which is called the connected edge cover polynomial of $ G $. After obtaining some results for this polynomial, we investigate this polynomial for some certain graphs.

math.CO↗

arXiv2024

Some resolving parameters with the minimum size for two specific graphs

A resolving set for a graph $G$ is a set of vertices $Q = \{q_1, ..., q_k\}$ such that, for all $p\in V(G)$ the $k$-tuple $(d(p, q_1), ..., d(p, q_k ))$ uniquely determines $p$, where $d(p, q_i)$ is considered as the minimum length of a shortest path from $p$ to $q_i$ in graph $G$. In this paper, we consider the computational study of some resolving sets with the minimum size for the $m$-cylinder graph $(C_n\Box P_k)\Box P_m$. The Boolean lattice $BL_n$, $n\geq 1$, is the graph whose vertex set is the set of all subsets of $[n]=\{1,2,...,n\}$, where two subsets $X$ and $Y$ are adjacent if their symmetric difference has precisely one element. In the graph $BL_n$, the layer $L_i$ is the family of $i$-subsets of $[n]$. The subgraph $BL_n(i,i+1)$ is the subgraph of $BL_n$ induced by layers $L_i$ and $L_{i+1}$. Usually the graph $BL_n(1,2)$ is denoted by $H(n)$. We study the minimum size of a resolving set, doubly resolving set and strong resolving set for the graph $L(n)$, which is the line graph of $H(n)$.

math.CO↗

arXiv2024

Strong coalitions in graphs

For a graph $G=(V,E)$, a set $D\subset V(G)$ is a strong dominating set of $G$, if for every vertex $x\in V (G)\setminus D$ there is a vertex $y\in D$ with $xy \in E(G)$ and $deg(x)\leq deg(y)$. A strong coalition consists of two disjoint sets of vertices $V_{1}$ and $V_{2}$, neither of which is a strong dominating set but whose union $V_{1}\cup V_{2}$, is a strong dominating set. A vertex partition $Ω=\{V_1, V_2,..., V_k \}$ of vertices in $G$ is a strong coalition partition, if every set $V_i \inΩ$ either is a strong dominating set consisting of a single vertex of degree $n-1$, or is not a strong dominating set but produces a strong coalition with another set $V_j \in Ω$ that is not a strong dominating set. The maximum cardinality of a strong coalition partition of $G$ is the strong coalition number of $G$ and is denoted by $SC(G)$. In this paper, we study properties of strong coalitions in graphs.

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