Transactions on Combinatorics
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Transactions on Combinatoricsendaily1Fri, 01 Dec 2023 00:00:00 +0330Fri, 01 Dec 2023 00:00:00 +0330Line graphs associated to annihilating-ideal graph attached to lattices of genus one
https://toc.ui.ac.ir/article_26897.html
Let $(L,\wedge,\vee)$ be a lattice with a least element $0$.&nbsp;The annihilating-ideal graph of $L$, denoted by&nbsp;$\mathbb{AG}(L)$, is a graph whose vertex-set is the set of all non-trivial ideals of $L$ and, for every&nbsp;two distinct vertices $I$ and $J$, the vertex $I$ is adjacent to $J$ if and only if $I\wedge J=\{0\}$. In this paper, we characterize all lattices $L$ whose the graph $\mathfrak{L}(\mathbb{AG}(L))$ is toroidal.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&nbsp;$V(G)=\{v_1, v_2,\ldots,v_n\}$ and edge set $E(G)$.&nbsp;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&nbsp;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.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 spectrum of generalized wheel graphs and a new class of graphs called dumbbell graphs.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 $d_{1}\geq d_{2}\geq \cdots d_{n}$ be the vertex degree sequence and $\mu_{1}\geq \mu_{2}\geq \cdots \geq \mu_{n-1}&gt;\mu_{n}=0$ 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.General sum-connectivity index of trees with given number of branching vertices
https://toc.ui.ac.ir/article_27009.html
In 2015, Borovi\'{c}anin presented trees with the smallest first Zagreb index among trees with given number of vertices and number of branching vertices. The first Zagreb index is obtained from the general sum-connectivity index if $a = 1$. For $a \in \mathbb{R}$, the general sum-connectivity index of a graph $G$ is defined as $\chi_{a} (G) = \sum_{uv\in E(G)} [d_G (u) + d_G (v)]^{a}$, where $E(G)$ is the edge set of $G$ and $d_G (v)$ is the degree of a vertex $v$ in $G$. We show that the result of Borovi\'{c}anin cannot be generalized for the general sum-connectivity index ($\chi_{a}$ index) if $0 &lt; a &lt; 1$ or $a &gt; 1$. Moreover, the sets of trees having the smallest $\chi_a$ index are not the same for $0 &lt; a &lt; 1$ and $a &gt; 1$. Among trees with given number of vertices and number of branching vertices, we present all the trees with the smallest $\chi_a$ index for $0 &lt; a &lt; 1$ and $a &gt; 1$. Since the hyper-Zagreb index is obtained from the $\chi_a$ index if $a = 2$, results on the hyper-Zagreb index are corollaries of our results on the $\chi_a$ index for $a &gt; 1$.The reformulated sombor index of a graph
https://toc.ui.ac.ir/article_27022.html
In 2021, Gutman invented a novel degree-based topological index called the Sombor index, inspired by a geometric interpretation of degree-radii of the edges and invited researchers to investigate their mathematical properties and chemical meanings. The Sombor index was reformulated in terms of the edge degree instead of the vertex degree as the original Sombor Index. In this paper, we compute the exact values of a certain class of graphs. Also, some bounds in terms of the order, size, minimum/maximum degrees and other topological indices are obtained.On graphs with anti-reciprocal eigenvalue property
https://toc.ui.ac.ir/article_27028.html
Let $\mathtt{A}(\mathtt{G})$ be the adjacency matrix of a simple connected undirected graph $\mathtt{G}$. A graph $\mathtt{G}$ of order $n$ is said to be non-singular (respectively singular) if $\mathtt{A}(\mathtt{G})$ is non-singular (respectively singular). The spectrum of a graph $\mathtt{G}$ is the set of all its eigenvalues denoted by $spec(\mathtt{G})$. The anti-reciprocal (respectively reciprocal) eigenvalue property for a graph $\mathtt{G}$ can be defined as `` Let $\mathtt{G}$ be a non-singular graph $\mathtt{G}$ if the negative reciprocal (respectively positive reciprocal) of each eigenvalue is likewise an eigenvalue of $\mathtt{G}$, then $\mathtt{G}$ has anti-reciprocal (respectively reciprocal) eigenvalue property ." Furthermore, a graph $\mathtt{G}$ is said to have strong anti-reciprocal eigenvalue property (resp. strong reciprocal eigenvalue property) if the eigenvalues and their negative (resp. positive) reciprocals are of same multiplicities. In this article, graphs satisfying anti-reciprocal eigenvalue (or property $(-\mathtt{R})$) and strong anti-reciprocal eigenvalue property (or property $(-\mathtt{SR})$) are discussed.Hadamard matrices of composite orders
https://toc.ui.ac.ir/article_27062.html
In this paper, we give a method for the constructions of Hadamard matrices of composite orders by using suitable $T$-matrices and known Hadamard matrices. We establish a formula for $T$-matrices and Hadamard matrices and discuss under what condition we can get $T$-matrices from the known Hadamard matrices.A spanning union of cycles in rectangular grid graphs, thick grid cylinders and Moebius strips
https://toc.ui.ac.ir/article_27132.html
Motivated to find the answers to some of the questions that have occurred in recent papers dealing with Hamiltonian cycles (abbreviated HCs) in some special classes of grid graphs we started the investigation of spanning unions of cycles, the so-called 2-factors, in these graphs (as a generalizations of HCs). For all the three types of graphs from the title and for any integer $m \geq 2$ we propose an algorithm for obtaining a specially designed (transfer) digraph ${\cal D}^*_m$. The problem of enumeration of 2-factors is reduced to the problem of enumerating oriented walks in this digraph. Computational results we gathered for $m \leq 17$ reveal some interesting properties both for the digraphs ${\cal D}^*_m$ and for the sequences of numbers of 2-factors.We prove some of them for arbitrary $m \geq 2$.Columns of fixed height in bargraphs
https://toc.ui.ac.ir/article_27194.html
We obtain the generating function for the number of columns of fixed height $r$ in a bargraph (classified according to semi-perimeter). As initial case for two distinct methods we first find the generating function for columns of height $1$. Then using a first-return-to-level-$1$ decomposition, we obtain the rational function version of the continued fraction generating function which allows us to derive separate recursions for its numerator and denominator. This then allows us to get the asymptotic average number of columns for each $r$. We also obtain an equivalent generating function by exploiting a sequential decomposition for bargraphs in terms of columns of height $r$.On variable sum exdeg energy of graphs
https://toc.ui.ac.ir/article_27250.html
In this paper, we put forward the idea of variable sum exdeg energy of graphs. We study the algebraic properties of variable sum exdeg energy. Some properties related to spectral radius of variable sum exdeg matrix are determined. We determine some Nordhaus-Gaddum-type results for variable sum exdeg spectral radius and energy. Some classes of variable sum exdeg equienergetic graphs are also determined.Comparing upper broadcast domination and boundary independence broadcast numbers of graphs
https://toc.ui.ac.ir/article_27258.html
A broadcast on a nontrivial connected graph $G=(V,E)$ is a function $f:V\rightarrow\{0, 1,\dots,d\}$, where $d=\operatorname{diam}(G)$, such that $f(v)\leq e(v)$ (the eccentricity of $v$) for all $v\in V$. The weight of $f$ is $\sigma(f)={\textstyle\sum_{v\in V}} f(v)$. A vertex $u$ hears $f$ from $v$ if $f(v)&gt;0$ and $d(u,v)\leq f(v)$. A broadcast $f$ is dominating if every vertex of $G$ hears $f$. The upper broadcast domination number of $G$ is $\Gamma_{b}(G)=\max\left\{ \sigma(f):f\text{ is a minimal dominating broadcast of }G\right\}.$&nbsp;A broadcast $f$ is boundary independent if, for any vertex $w$ that hears $f$ from vertices $v_{1},\ldots,v_{k},\ k\geq2$, the distance $d(w,v_{i})=f(v_{i})$ for each $i$. The maximum weight of a boundary independent broadcast is the boundary independence broadcast number $\alpha_{\operatorname{bn}}(G)$.&nbsp;We compare $\alpha_{\operatorname{bn}}$ to $\Gamma_{b}$, showing that neither is an upper bound for the other. We show that the differences $\Gamma _{b}-\alpha_{\operatorname{bn}}$ and $\alpha_{\operatorname{bn}}-\Gamma_{b}$ are unbounded, the ratio $\alpha_{\operatorname{bn}}/\Gamma_{b}$ is bounded for all graphs, and $\Gamma_{b}/\alpha_{\operatorname{bn}}$ is bounded for bipartite graphs but unbounded in general.A new $q$-analogue of the binomial identity $\sum_{k}(-1)^k{2n\choose n+3k}=2\cdot 3^{n-1}$
https://toc.ui.ac.ir/article_27269.html
In this paper, we establish a new $q$-analogue of the binomial identity:
\begin{align*}
&amp;\sum_{k}(-1)^k{2n\choose n+3k}=
\begin{cases}
1,&amp;\text{if $n=0$,}\\[5pt]
2\cdot3^{n-1},&amp;\text{if $n\ge 1$.}
\end{cases}
\end{align*}
Our proof relies on a weight-preserving and sign-reversing involution due to Guo and Zhang.On the skew spectral moments of trees with a given bipartition
https://toc.ui.ac.ir/article_27275.html
Let $G$ be a simple graph, and $\vec{G}$ be an oriented graph of $G$ with an orientation and skew-adjacency matrix $S(\vec{G})$. Let $\lambda_1(\vec{G}), \lambda_2(\vec{G}),\ldots,\lambda_n(\vec{G})$ be the eigenvalues of $S(\vec{G})$. The number $\sum_{i=1}^{n}\lambda_i^k(\vec{G})$ $(k=0, 1,\ldots,n-1)$, denoted by $T_k(\vec{G})$, is called the $k$-th {\em skew spectral moment} of $\vec{G}$, and $T(\vec{G})=(T_0(\vec{G}),T_1(\vec{G}),\ldots,$ $T_{n-1}(\vec{G}))$ is the sequence of skew spectral moments of $\vec{G}$. Suppose $\vec{G}_1$ and $\vec{G}_2$ are two digraphs. We shall write $\vec{G}_1\prec_T \vec{G}_2$ ($\vec{G}_1$ comes before $\vec{G}_2$ in a $T$-order) if for some $k$ $(1 \leq k \leq n-1)$, $T_i(\vec{G}_1)=T_i(\vec{G}_2)$ ($i=0, 1,\ldots,k-1$) and $T_k(\vec{G_1})&lt; T_k(\vec{G}_2)$ hold. For two given positive integers $p$ and $q$ with $p \leq q$, we denote $\mathscr T_{n}^{p,q}=\{T: T$ is a tree of order $n$ with a $(p,q)$-bipartition $\}$. In this paper, we discuss $T$-order among all trees in $\mathscr T_{n}^{p,q}$. Furthermore, the last three trees, in the $T$-order, underlying graphs among $\mathscr T_{n}^{p,q}~(4\leq p\leq q)$ are characterized.Some results on non-progressive spread of influence in graphs
https://toc.ui.ac.ir/article_27387.html
This paper studies the non-progressive spread of influence with threshold model in social networks. Consider a graph $G$ with a threshold function $\tau$ on its vertex set. Spread of influence is a discrete dynamic process as follows. For a given set of initially infected vertices at time step $0$ each vertex $v$ gets infected at time step $i$, $i\geq1$, if and only if the number of infected neighbors are at least $\tau(v)$ in time step $i-1$. Our goal is to find the minimum cardinality of initially infected vertices (perfect target set) such that eventually all of the vertices get infected at some time step $\ell$.
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In this paper an upper bound for the convergence time of the process is given for graphs with general thresholds. Then an integer linear programming for the size of minimum perfect target set is presented. Then we give a lower bound for the size of perfect target sets with strict majority threshold on graphs which all of the vertices have even degrees. It is shown that the later bound is asymptotically tight.The Higman-Sims sporadic simple group as the automorphism group of resolvable $3$-designs
https://toc.ui.ac.ir/article_27449.html
Presenting sporadic simple groups as an automorphism groups of designs and graphs is an exciting field in finite group theory.
In this paper, with two different methods, we present some new resolvable simple $3$-designs with Higman-Sims sporadic simple group $\rm HS$ as the full automorphism group.
Also, we classify all block-transitive self-orthogonal designs on 176 points with even block size that admit sporadic simple group $\rm HS$ as an automorphism group. Furthermore, with these methods we construct some new resolvable $3$-designs on 36, 40, 120 and 176 points.Graphs without a $2C_3$-minor and bicircular\\ matroids without a $U_{3,6}$-minor
https://toc.ui.ac.ir/article_27470.html
In this note we characterize all graphs without a $2C_3$-minor. A consequence of this result is a characterization of the bicircular matroids with no $U_{3,6}$-minor.On topological charge indices of graphs
https://toc.ui.ac.ir/article_27517.html
We introduce a fast method of computing the topological charge indices of simple graphs (molecules) which does not require matrices of large sizes. For the case of trees, we give a compact formula and in the general case we obtain upper and lower bounds for the charge indices. We give concrete examples of trees and molecules with their charge indices computed using our method.Whitney numbers of partial dowling lattices
https://toc.ui.ac.ir/article_27575.html
The Dowling lattice $Q_n(G)$, $G$ a finite group, generalizes the geometric lattice generated by all vectors, over a field, with at most two nonzero components. Abstractly, it is a fundamental object in the classification of finite matroids. Constructively, it is the frame matroid of a certain gain graph known as $G K{_n}{^V}$. Its Whitney numbers of the first kind enter into several important formulas. Ravagnani suggested and partially proved that these numbers of $Q_n(G)$ and higher-weight generalizations are polynomial functions of $|G|$. We give a simple proof for $Q_n(G)$ and its generalization to a wider class of gain graphs and biased graphs, and we determine the degrees and coefficients of the polynomials.