Transactions on Combinatorics
https://toc.ui.ac.ir/
Transactions on Combinatoricsendaily1Thu, 01 Dec 2022 00:00:00 +0330Thu, 01 Dec 2022 00:00:00 +0330Total 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 $|Z^*(R)|$ number of 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.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 $\pi(G\square H)$ 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 $\pi(G\square H)$ are the pebbling number of graphs $G$, $H$ and the Cartesian product graph $G\square H$, respectively. Moreover, we also discuss such bound for strong product graphs, cross product graphs and coronas.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 $\chi(G)$ be the chromatic number of $G$&nbsp;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 $q(G)=2n\Big(1-\frac{1}{\chi}\Big)$. Finally we show that if $G$ is a graph of order $n$ and with chromatic number $\chi$, then under certain conditions, $q(G)&lt;2n\Big(1-\frac{1}{\chi}\Big)-\frac{2}{n}$. This result improves some previous similar results.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.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.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 $O(\log^{d+1}n)$.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.Domination number of middle graphs
https://toc.ui.ac.ir/article_26638.html
In this paper, we study the domination number of middle graphs. Indeed, we obtain tight bounds for this number in terms of the order of the graph G. We also compute the domination number of some families of graphs such as star graphs, double start graphs, path graphs, cycle graphs, wheel graphs, complete graphs, complete bipartite graphs and friendship graphs, explicitly. Moreover, some Nordhaus-Gaddum-like relations are presented for the domination number of middle graphs.On the Zagreb index of random m-oriented recursive trees
https://toc.ui.ac.ir/article_26761.html
The main goal of this paper is to study the modified $F$-indices (modified first Zagreb index and modified forgotten topological index) of&nbsp;random $m$-oriented recursive trees (RMORTs).&nbsp;First, through two recurrence equations, we compute the mean and the variance of these indices in our random tree model. Second, we show four convergence&nbsp;in probability based on these indices.&nbsp;Third, the asymptotic normalities, through the martingale central limit theorem,&nbsp;are given.Semi Square Stable Graphs and Efficient Dominating Sets
https://toc.ui.ac.ir/article_26787.html
A graph $G$ is called semi square stable if $\alpha (G^{2})=i(G)$ where $%\alpha (G^{2})$ is the independence number of $G^{2}$ and $i(G)$ is the&nbsp;independent dominating number of $G$. A subset $S$ of the vertex set of a&nbsp;graph $G$ is an efficient dominating set if $S$ is an independent set and&nbsp;every vertex of $G$ is either in $S$ or adjacent to exactly one vertex of $%S. $
In this paper, we show that every square stable graph has an efficient&nbsp;dominating set and if a graph has an efficient dominating set, then it is&nbsp;semi square stable. We characterize when the join and the corona product of&nbsp;two disjoint graphs are semi square sable graphs and when they have&nbsp;efficient dominating sets.&nbsp;The number of graph homomorphisms between paths and cycles with loops, a problem from stanleyâ€™s enumerative combinatorics
https://toc.ui.ac.ir/article_26788.html
Let $g_{k}(n)$ denote the number of sequences $t_{1},ldots,t_{n}$ in $\{0, 1,\ldots,k-1\}$ such that $t_{j+1}\equiv t_{j}-1, t_{j}$ or $t_{j}+1$ (mod $k$), $1\leq j\leq n$, (where $t_{n+1}$ is identified with $t_{1}$). It is proved combinatorially that $g_{4}(n)= 3^{n}+2+(-1)^{n}$ and $g_{6}(n)= 3^{n}+2^{n+1}+(-1)^{n}$. This solves a problem from Richard P. Stanley's 1986 text, $Enumerative$ $Combinatorics$.The normalized signless laplacian estrada index of graphs
https://toc.ui.ac.ir/article_26789.html
Let $G$ be a simple connected graph of order $n$ with $m$ edges. Denote by $%&nbsp;\gamma _{1}^{+}\geq \gamma _{2}^{+}\geq \cdots \geq \gamma _{n}^{+}\geq 0$&nbsp;the normalized signless Laplacian eigenvalues of $G$. In this work, we&nbsp;define the normalized signless Laplacian Estrada index of $G$ as $NSEE\left(G\right) =\sum_{i=1}^{n}e^{\gamma _{i}^{+}}.$ Some lower bounds on $%NSEE\left( G\right) $ are also established.Transformations among rectangular partitions
https://toc.ui.ac.ir/article_26808.html
We first prove that there always exists a maximal rectangularly dualizable graph for a given rectangularly dualizable graph and present an algorithm for its construction. Further, we show that a maximal rectangularly dualizable graph can always be transformed to an edge-irreducible rectangularly dualizable graph and present an algorithm that transforms a maximal rectangularly dualizable graph to an edge-irreducible rectangularly dualizable graph.Energy of strong reciprocal graphs
https://toc.ui.ac.ir/article_26810.html
The energy of a graph $G$, denoted by $\mathcal{E}(G)$, is defined as the sum of absolute values of all eigenvalues of $G$.&nbsp;A graph $G$ is called reciprocal if $ \frac{1}{\lambda} $ is an eigenvalue of $G$ whenever $\lambda$ is an eigenvalue of $G$. Further, if $ \lambda $ and $\frac{1}{\lambda}$ have the same multiplicities, for each eigenvalue $\lambda$, then it is called strong reciprocal. In (MATCH Commun. Math. Comput. Chem. 83 (2020) 631--633), it was conjectured that for every graph $G$ with maximum degree $\Delta(G)$ and minimum degree $\delta(G)$ whose adjacency matrix is non-singular, $\mathcal{E}(G) \geq \Delta(G) + \delta(G)$ and the equality holds if and only if $G$ is a complete graph. Here,&nbsp;we prove the validity of this conjecture for some strong reciprocal graphs. Moreover, we show that if $G$ is a strong reciprocal graph, then&nbsp;$\mathcal{E}(G) \geq \Delta(G) + \delta(G) - \frac{1}{2}$. Recently, it has been proved that if $G$ is a reciprocal graph of order $n$ and its spectral radius, $\rho$, is at least $4\lambda_{min}$, where $ \lambda_{min}$ is the smallest absolute value of eigenvalues of $G$, then $\mathcal{E}(G) \geq n+\frac{1}{2}$. In this paper, we extend this result to almost all strong reciprocal graphs without the mentioned assumption.Counterexamples to a conjecture on matching Kneser graphs
https://toc.ui.ac.ir/article_26847.html
Let $G$ be a graph and $r\in\mathbb{N}$. The matching Kneser graph $\textsf{KG}(G, rK_2)$ is a graph whose vertex set is the set of $r$-matchings in $G$ and two vertices are adjacent if their corresponding matchings are edge-disjoint. In [M. Alishahi and H. Hajiabolhassan, On the Chromatic Number of Matching Kneser Graphs, Combin. Probab. and Comput., 29, No. 1 (2020), 1--21] it was conjectured that for any connected graph $G$ and positive integer $r\geq 2$, the chromatic number of $\textsf{KG}(G, rK_2)$ is equal to $|E(G)|-\textsf{ex}(G,rK_2)$, where $\textsf{ex}(G,rK_2)$ denotes the largest number of edges in $G$ avoiding a matching of size $r$. In this note, we show that the conjecture is not true for snarks.On the spectral radius, energy and Estrada index of the Sombor matrix of graphs
https://toc.ui.ac.ir/article_26896.html
Let $G$ be a simple undirected graph with vertex set $V(G)=\{v_1, v_2, \ldots, v_n\}$ and edge set $E(G)$.The Sombor matrix $\mathcal{S}(G)$ of a graph $G$ is defined so that its $(i,j)$-entry is equal to $\sqrt{d_i^2+d_j^2}$ if the vertices $v_i$ and $v_j$ are adjacent, and zero otherwise, where $d_i$ denotes the degree of vertex $v_i$ in $G$. In this paper, lower and upper bounds on the spectral radius, energy and Estrada index of the Sombor matrix of graphs are obtained, and the respective extremal graphs are characterized.Line graphs associated to annihilating-ideal graph attached to lattices of genus one
https://toc.ui.ac.ir/article_26897.html
&lrm;Let $(L,\wedge,\vee)$ be a lattice with a least element $0$&lrm;.&lrm;The annihilating-ideal graph of $L$&lrm;, &lrm;denoted by&lrm;&lrm;$\AG(L)$&lrm;, &lrm;is a graph whose vertex-set is the set of all non-trivial ideals of $L$ and&lrm;, &lrm;for every&lrm;&lrm;two distinct vertices $I$ and $J$&lrm;, &lrm;the vertex $I$ is adjacent to $J$ if and only if $I\wedge J=\{0\}$&lrm;. &lrm;In this paper&lrm;, &lrm;we characterize all lattices $L$ whose the graph $\mathfrak{L}(\AG(L))$ is toroidal&lrm;.On Laplacian resolvent energy of graphs
https://toc.ui.ac.ir/article_26922.html
Let $G$ be a simple connected graph of order $n$ and size $m$. The matrix $L(G)=D(G)-A(G)$ is the Laplacian matrix of $G$, where $D(G)$ and $A(G)$ are the degree diagonal matrix and the adjacency matrix, respectively. For the graph $G$, let $d_{1}\geq d_{2}\geq \dots d_{n}$ be the vertex degree sequence and $\mu_{1}\geq \mu_{2}\geq \dots \geq \mu_{n-1}&gt;\mu_{n}=0$ be the Laplacian eigenvalues. The Laplacian resolvent energy $RL(G)$ of a graph $G$ is defined as $RL(G)=\sum\limits_{i=1}^{n}\frac{1}{n+1-\mu_{i}}$. In this paper, we obtain an upper bound for the Laplacian resolvent energy $RL(G)$ in terms of the order, size and the algebraic connectivity of the graph. Further, we establish relations between the Laplacian resolvent energy $RL(G)$ with each of the Laplacian-energy-Like invariant $LEL$, the Kirchhoff index $Kf$ and the Laplacian energy $LE$ of the graph.DISTANCE (SIGNLESS) LAPLACIAN SPECTRUM OF DUMBBELL GRAPHS
https://toc.ui.ac.ir/article_26923.html
In this paper, we determine the distance Laplacian and distance signless Laplacian spectrumof generalized wheel graphs and a new class of graphs called dumbbell graphs.