Transactions on Combinatorics
https://toc.ui.ac.ir/
Transactions on Combinatoricsendaily1Wed, 01 Jun 2022 00:00:00 +0430Wed, 01 Jun 2022 00:00:00 +0430Peripheral Hosoya polynomial of composite graphs
https://toc.ui.ac.ir/article_26025.html
Peripheral Hosoya polynomial of a graph $G$ is defined as&lrm;,&lrm; \begin{align*}&lrm;&lrm; &amp;PH(G,\lambda)=\sum_{k\geq 1}d_P(G,k)\lambda^k,\\&lrm;&lrm; \text{where $d_P(G,k)$ is the number} &amp;\text{ of pairs of peripheral vertices at distance $k$ in $G$.}&lrm;&lrm; \end{align*}&lrm;Peripheral Hosoya polynomial of composite graphs viz.&lrm;, &lrm;$G_1\times G_2$ the Cartesian product&lrm;, &lrm;$G_1+G_2$ the join&lrm;, &lrm;$G_1[G_2]$ the composition&lrm;, &lrm;$G_1\circ G_2$ the corona and $G_1\{G_2\}$ the cluster of arbitrary connected graphs $G_1$ and $G_2$ are computed and their peripheral Wiener indices are stated as immediate consequences&lrm;.Bounds for the pebbling number of product graphs
https://toc.ui.ac.ir/article_25997.html
Let $G$ be a connected graph. Given a configuration of a fixed number of pebbles on the vertex set of $G$, a pebbling move on $G$ is the process of removing two pebbles from a vertex and adding one pebble on an adjacent vertex. The pebbling number of $G$, denoted by $&pi;(G)$, is defined to be the least number of pebbles to guarantee that there is a sequence of pebbling movement that places at least one pebble on each vertex $v$, for any configuration of pebbles on $G$. In this paper, we improve the upper bound of $&pi;(G◻H)$ from $2&pi;(G)&pi;(H)$ to $\left(2-\frac{1}{\min\{&pi;(G),&pi;(H)\}}\right)&pi;(G)&pi;(H)$ where $&pi;(G)$, $&pi;(H)$ and $&pi;(G◻H)$ are the pebbling number of graphs $G$, $H$ and the Cartesian product graph $G◻H$, respectively. Moreover, we also discuss such bound for strong product graphs, cross product graphs and coronas.On the reliability of modified bubble-sort graphs
https://toc.ui.ac.ir/article_25913.html
The modified bubble-sort graph $MB_n$ $(n \geq 2)$ has been known as a topology structure of interconnection networks&lrm;. &lrm;In this paper&lrm;, &lrm;we propose simple method for arc-transitivity of $MB_n$ $(n \geq 2)$&lrm;. &lrm;Also by using this result we investigate some reliability measures&lrm;, &lrm;including&lrm;&nbsp;&lrm;super-connectivity&lrm;, &lrm;cyclic edge connectivity&lrm;, &lrm;etc.&lrm;, &lrm;in the modified bubble-sort graphs&lrm;.On the characteristic polynomial and spectrum of Basilica Schreier graphs
https://toc.ui.ac.ir/article_26080.html
The Basilica group is one of the most studied automaton groups, and many papers have been devoted to the investigation of the characteristic polynomial and spectrum of the associated Schreier graphs $\{\Gamma_n\}_{n\geq 1}$, even if an explicit description of them has not been given yet. Our approach to this issue is original, and it is based on the use of the Coefficient Theorem for signed graphs. We introduce a signed version $\Gamma_n^-$ of the Basilica Schreier graph $\Gamma_n$, and we prove that there exist two fundamental relations between the characteristic polynomials of the signed and unsigned versions. The first relation comes from the cover theory of signed graphs. The second relation is obtained by providing a suitable decomposition of $\Gamma_n$ into three parts, using the self-similarity of $\Gamma_n$, via a detailed investigation of its basic figures. By gluing together these relations, we find out a new recursive equation which expresses the characteristic polynomial of $\Gamma_n$ as a function of the characteristic polynomials of the three previous levels. We are also able to give an explicit description of the eigenspace associated with the eigenvalue $2$, and to determine how the eigenvalues are distributed with respect to such eigenvalue.Total perfect codes in graphs realized by commutative rings
https://toc.ui.ac.ir/article_26081.html
Let $R$ be a commutative ring with unity not equal to zero and let $\Gamma(R)$ be a zero-divisor graph realized by $R$. For a simple, undirected, connected graph $G = (V, E)$, a {\it total perfect code} denoted by $C(G)$ in $G$ is a subset $C(G) \subseteq V(G)$ such that $|N(v) \cap C(G)| = 1$ for all $v \in V(G)$, where $N(v)$ denotes the open neighbourhood of a vertex $v$ in $G$. In this paper, we study total perfect codes in graphs which are realized as zero-divisor graphs. We show a zero-divisor graph realized by a local commutative ring with unity admits a total perfect code if and only if the graph has degree one vertices. We also show that if $\Gamma(R)$ is a regular graph on $|Z^*(R)|$ vertices, then $R$ is a reduced ring and $|Z^*(R)| \equiv 0(mod ~2)$, where $Z^*(R)$ is a set of non-zero zero-divisors of $R$. We provide a characterization for all commutative rings with unity of which the realized zero-divisor graphs admit total perfect codes. Finally, we determine the cardinality of a total perfect code in $\Gamma(R)$ and discuss the significance of the study of total perfect codes in graphs realized by commutative rings with unity.The identifying code number and Mycielski's construction of graphs
https://toc.ui.ac.ir/article_26088.html
Let $G=(V, E)$ be a simple graph. A set $C$ of vertices $G$ is an identifying code of $G$ if for every two vertices $x$ and $y$ the sets $N_{G} [x] \cap C$ and $N_{G} [y] \cap C$ are non-empty and different. Given a graph $G,$ the smallest size of an identifying code of $G$ is called the identifying code number of $G$ and denoted by $\gamma^{ID}(G).$ Two vertices $x$ and $y$ are twins when $N_{G}[x]=N_{G}[y].$ Graphs with at least two twin vertices are not an identifiable graph. In this paper, we deal with the identifying code number of Mycielski's construction of graph $G.$ We prove that the Mycielski's construction of every graph $G$ of order $n \geq 2,$ is an identifiable graph. Also, we present two upper bounds for the identifying code number of Mycielski's construction $G,$ such that these two bounds are sharp. Finally, we show that Foucaud et al.'s conjecture is holding for Mycielski's construction of some graphs.A list of applications of Stallings automata
https://toc.ui.ac.ir/article_26120.html
This survey is intended to be a fast (and reasonably updated) reference for the theory of Stallings automata and its applications to the study of subgroups of the free group, with the main accent on algorithmic aspects. Consequently, results concerning finitely generated subgroups have greater prominence in the paper. However, when possible, we try to state the results with more generality, including the usually overlooked non-(finitely-generated) case.Chromatic number and signless Laplacian spectral radius of graphs
https://toc.ui.ac.ir/article_26159.html
For any simple graph $G$, the signless Laplacian matrix of $G$ is deﬁned as $D(G) + A(G)$, where $D(G)$ and $A(G)$ are the diagonal matrix of vertex degrees and the adjacency matrix of $G$, respectively. Let $q(G)$ be the signless Laplacian spectral radius of $G$ (the largest eigenvalue of the signless Laplacian matrix of $G$). In this paper we ﬁnd some relations between the chromatic number and the signless Laplacian spectral radius of graphs. In particular, we characterize all graphs $G$ of order $n$ with odd chromatic number $&chi;$ such that $q(G) = 2n1&minus; 1 &chi;$. Finallywe show that if $G$ is a graph of order $n$ and with chromatic number $&chi;$, then under certain conditions, $q(G) &lt; 2n1&minus; 1 &chi;&minus; 2 n$. This result improves some previous similar results.On eigenspaces of some compound complex unit gain graphs
https://toc.ui.ac.ir/article_26186.html
Let $\mathbb T$ be the multiplicative group of complex units, and let $L(\Phi)$ denote the Laplacian matrix of a nonempty $\mathbb{T}$-gain graph $\Phi=(\Gamma, \mathbb{T}, \gamma)$. The gain line graph $\mathcal L(\Phi)$ and the gain subdivision graph $\mathcal S(\Phi)$ are defined up to switching equivalence. We discuss how the eigenspaces determined by the adjacency eigenvalues of $\mathcal L(\Phi)$ and $\mathcal S(\Phi)$ are related with those of $L(\Phi)$Linear codes resulting from finite group actions
https://toc.ui.ac.ir/article_26249.html
In this article, we use group action theory to define some important ternary linear codes. Some of these codes are self-orthogonal having a minimum distance achieving the lower bound in the previous records. Then, we define two new codes sharing the same automorphism group isomorphic to $C_2 \times M_{11}$ where $M_{11}$ is the Sporadic Mathieu group and $C_{2}$ is a cyclic group of two elements. We also study the natural action of the general linear group $GL (k, 2) $ on the vector space $F_2 ^ k$ to characterize Hamming codes $H_k (2) $ and their automorphism group.Maximum second Zagreb index of trees with given roman domination number
https://toc.ui.ac.ir/article_26290.html
Chemical study regarding total $\pi$-electron energy with respect to conjugated molecules has focused on the second Zagreb index of graphs. Moreover, in the last half-century, it has gotten a lot of attention. The relationship between the Roman domination number and the second Zagreb index is investigated in this study. We characterize the trees with the maximum second Zagreb index among those with the given Roman domination number.Conditional probability of derangements and fixed points
https://toc.ui.ac.ir/article_26291.html
The probability that a random permutation in $S_n$ is a derangement is well known to be $\displaystyle\sum\limits_{j=0}^n (-1)^j \frac{1}{j!}$. In this paper,&nbsp;we consider the conditional probability that the $(k+1)^{st}$ point is fixed, given there are no fixed points in the first $k$ points.&nbsp;We prove that when $n \neq 3$ and $k \neq 1$, this probability is a decreasing function of both $k$ and $n$. Furthermore, it is proved that this conditional probability is well approximated by $\frac{1}{n} - \frac{k}{n^2(n-1)}$. Similar results are also obtained about the more general conditional probability that the $(k+1)^{st}$ point is fixed, given that there are exactly $d$ fixed points in the first $k$ pointsSome chemical indices related to the number of triangles
https://toc.ui.ac.ir/article_26292.html
Many chemical indices have been invented in theoretical chemistry, such as the Zagreb index, the Lanzhou index, the forgotten index, the Estrada index etc. In this paper, we show that the first Zagreb index is only related to the sum of the number of triangles in a graph and the number of triangles in its complement. Moreover, we determine the sum of the first and second Zagreb index, the Lanzhou index and the forgotten index for a graph and its complement in terms of the number of triangles in a graph and the number of triangles in its complement. Finally, we estimate the Estrada index in terms of order, size, maximum degree and the number of triangles.The Mostar and Wiener index of alternate Lucas cubes
https://toc.ui.ac.ir/article_26293.html
The Wiener index and the Mostar index quantify two distance related properties of connected graphs: the Wiener index is the sum of the distances over all pairs of vertices and the Mostar index is a measure of how far the graph is from being distance-balanced.&nbsp;These two measures have been considered for a number of interesting families of graphs.&nbsp;In this paper, we determine the Wiener index and the Mostar index of alternate Lucas cubes. Alternate Lucas cubes form a family of interconnection networks whose recursive construction mimics the construction of the well-known Fibonacci cubes.A directed graph associated with a $T_0$-quasi-metric space
https://toc.ui.ac.ir/article_26320.html
Given a $T_0$-quasi-metric space we associate a directed graph with it and study some properties of the related directed graph.&nbsp;The present work complements and refines earlier work in the field in which the symmetry graph of a $T_0$-quasi-metric space was studied.Generalized Barred Preferential Arrangements
https://toc.ui.ac.ir/article_26337.html
We investigate a generalization of Fubini numbers. We present the combinatorial interpretation as barred preferential arrangements with some additional conditions on the blocks. We provide a proof for a generalization of Nelsen's Theorem. We consider these numbers from a probabilistic view point and demonstrate how they can be written in terms of the expectation of random descending factorial involving the negative binomial process.A short note on the topological decomposition of the Central Product of Groups
https://toc.ui.ac.ir/article_26350.html
It has been recently observed that a topological decomposition of the Pauli group, as central product of the quaternion group of order eight and the cyclic group of order four, influences some significant dynamical systems in mathematical physics. The connection between groups of symmetries and dynamical systems is in fact well known, but looking specifically at the algebraic and topological decompositions of the Pauli group, we find conditions for the existence of a Riemannian $3$-manifold whose fundamental group is epimorphically mapped onto a central product.Approximate $k$-Nearest Neighbor Graph on Moving Points
https://toc.ui.ac.ir/article_26356.html
In this paper, we introduce an approximation for the $k$-nearest neighbor graph ($k$-NNG) on a point set $P$ in $\mathbb{R}^d$. For any given $\varepsilon&gt;0$, we construct a graph, that we call the \emph{approximate $k$-NNG}, where the edge with the $i$th smallest length incident to a point $p$ in this graph is within a factor of $(1+\varepsilon)$ of the length of the edge with the $i$th smallest length incident to $p$ in the $k$-NNG.
&nbsp;
For a set $P$ of $n$ moving points in $\mathbb{R}^d$, where the trajectory of each point $p\in P$ is given by $d$ polynomial functions of constant bounded degree, where each function gives one of the $d$ coordinates of $p$, we compute the number of combinatorial changes to the approximate $k$-NNG, and provide a kinetic data structure (KDS) for maintenance of the edges of the approximate $k$-NNG over time. Our KDS processes $O(kn^2\log^{d+1} n)$ events, each in time $O(\log^{d+1}n)$.A polynomial associated with rooted trees and specific posets
https://toc.ui.ac.ir/article_26419.html
We investigate a trivariate polynomial associated with rooted trees. It generalises a bivariate polynomial for rooted trees that was recently introduced by Liu. We show that this polynomial satisfies a deletion-contraction recursion and can be expressed as a sum over maximal antichains. Several combinatorial quantities can be obtained as special values, in particular the number of antichains, maximal antichains and cutsets. We prove that two of the three possible bivariate specialisations characterise trees uniquely up to isomorphism. One of these has already been established by Liu, the other is new. For the third specialisation, we construct non-isomorphic trees with the same associated polynomial. We finally find that our polynomial can be generalised in a natural way to a family of posets that we call V-posets. These posets are obtained recursively by either disjoint unions or adding a greatest/least element to existing V-posets.Spectral properties of the non--permutability graph of subgroups
https://toc.ui.ac.ir/article_26482.html
Given a finite group $G$ and the subgroups lattice $\mathrm{L}(G)$ of $G$, the non--permutability graph of subgroups $\Gamma_{\mathrm{L}(G)}$ is introduced as the graph with vertices in $\mathrm{L}(G) \setminus \mathfrak{C}_{\mathrm{L}(G)}(\mathrm{L}(G))$, where $\mathfrak{C}_{\mathrm{L}(G)}(\mathrm{L}(G))$ is the smallest sublattice of $\mathrm{L}(G)$ containing all permutable subgroups of $G$, and edges obtained by joining two vertices $X,Y$ if $XY\neq YX$. Here we study the behaviour of the non-permutability graph of subgroups using algebraic properties of associated matrices such as the adjacency and the Laplacian matrix. Further, we study the structure of some classes of groups whose non-permutability graph is strongly regular.Unicyclic graphs with non-isolated resolving number $2$
https://toc.ui.ac.ir/article_26496.html
Let $G$ be a connected graph and $W=\{w_1, w_2,\ldots,w_k\}$ be an ordered subset of vertices of $G$. For any vertex $v$ of $G$, the ordered $k$-vector $$r(v|W)=(d(v,w_1), d(v,w_2),\ldots,d(v,w_k))$$ is called the metric representation of $v$ with respect to $W$, where $d(x,y)$ is the distance between the vertices $x$ and $y$. A set $W$ is called a resolving set for $G$ if distinct vertices of $G$ have distinct metric representations with respect to $W$. The minimum cardinality of a resolving set for $G$ is its metric dimension denoted by $\dim(G)$. A resolving set $W$ is called a non-isolated resolving set for $G$ if the induced subgraph $\langle W\rangle$ of $G$ has no isolated vertices. The minimum cardinality of a non-isolated resolving set for $G$ is called the non-isolated resolving number of $G$ and denoted by $nr(G)$. The aim of this paper is to find properties of unicyclic graphs that have non-isolated resolving number $2$ and then to characterize all these graphs.Optimal maximal graphs
https://toc.ui.ac.ir/article_26015.html
An optimal labeling of a graph with $n$ vertices and $m$ edges is an injective assignment of the first $n$ nonnegative integers to the vertices&lrm;, &lrm;that induces&lrm;, &lrm;for each edge&lrm;, &lrm;a weight given by the sum of the labels of its end-vertices with the property that the set of all induced weights consists of the first $m$ positive integers&lrm;. &lrm;We explore the connection of this labeling with other well-known functions such as super edge-magic and $\alpha$-labelings&lrm;. &lrm;A graph with $n$ vertices is maximal when the number of edges is $2n-3$; all the results included in this work are about maximal graphs&lrm;. &lrm;We determine the number of optimally labeled graphs using the adjacency matrix&lrm;. &lrm;Several techniques to construct maximal graphs that admit an optimal labeling are introduced as well as a family of outerplanar graphs that can be labeled in this form.Independent roman $\{3\}$-domination
https://toc.ui.ac.ir/article_26066.html
Let $G$ be a simple, undirected graph. In this paper, we initiate the study of independent Roman $\{3\}$-domination. A function $g : V(G) \rightarrow \lbrace 0, 1, 2, 3 \rbrace$ having the property that $\sum_{v \in N_G(u)}^{} g(v) \geq 3$, if $g(u) = 0$, and $\sum_{v \in N_G(u)}^{} g(v) \geq 2$, if $g(u) = 1$ for any vertex $u \in V(G)$, where $N_G(u)$ is the set of vertices adjacent to $u$ in $G$, and no two vertices assigned positive values are adjacent is called an \textit{ independent Roman $\{3\}$-dominating function} (IR3DF) of $G$.&nbsp;The weight of an IR3DF $g$ is the sum $g(V) = \sum_{v \in V}g(v)$.&nbsp;Given a graph $G$ and a positive integer $k$, the independent Roman $\{3\}$-domination problem (IR3DP) is to check whether $G$ has an IR3DF of weight at most $k$.&nbsp;We investigate the complexity of IR3DP in bipartite and chordal graphs.&nbsp;The minimum independent Roman $\lbrace 3 \rbrace$-domination problem (MIR3DP) is to find an IR3DF of minimum weight in the input graph.&nbsp;We show that MIR3DP is linear time solvable for bounded tree-width graphs, chain graphs and threshold graphs.&nbsp;We also show that the domination problem and IR3DP are not equivalent in computational complexity aspects. Finally, we present an integer linear programming formulation for MIR3DP.Nordhaus-gaddum type inequalities for tree covering numbers on unitary cayley graphs of finite rings
https://toc.ui.ac.ir/article_26053.html
The unitary Cayley graph $\Gamma_n$ of a finite ring $\mathbb{Z}_n$ is the graph with vertex set $\mathbb{Z}_n$ and two vertices $x$ and $y$ are adjacent if and only if $x-y$ is a unit in $\mathbb{Z}_n$&lrm;. &lrm;A family $\mathcal{F}$ of mutually edge disjoint trees in $\Gamma_n$ is called a tree cover of $\Gamma_n$ if for each edge $e\in E(\Gamma_n)$&lrm;, &lrm;there exists a tree $T\in\mathcal{F}$ in which $e\in E(T)$&lrm;. &lrm;The minimum cardinality among tree covers of $\Gamma_n$ is called a tree covering number and denoted by $\tau(\Gamma_n)$&lrm;. &lrm;In this paper&lrm;, &lrm;we prove that&lrm;, &lrm;for a positive integer $ n\geq 3 $&lrm;, &lrm;the tree covering number of $ \Gamma_n $ is $ \displaystyle\frac{\varphi(n)}{2}+1 $ and the tree covering number of $ \overline{\Gamma}_n $ is at most $ n-p $ where $ p $ is the least prime divisor of $n$&lrm;. &lrm;Furthermore&lrm;, &lrm;we introduce the Nordhaus-Gaddum type inequalities for tree covering numbers on unitary Cayley graphs of rings $\mathbb{Z}_n$&lrm;.