Transactions on Combinatorics
https://toc.ui.ac.ir/
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 $\ell$.
&nbsp;
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.Total Roman domination and $2$-independence in trees
https://toc.ui.ac.ir/article_27601.html
Let $G=(V, E)$ be a simple graph with vertex set $V$ and edge set $E$. A {\em total Roman dominating function} on a graph $G$ is a function $f:V\rightarrow \{0,1,2\}$ satisfying the following conditions: (i) every vertex $u$ {\color{blue}such that} $f(u)=0$ is adjacent to at least one vertex $v$ {\color{blue}such that} $f(v)=2$ and (ii) the subgraph of $G$ induced by the set of all vertices of positive weight has no isolated vertex. The weight of a total Roman dominating function $f$ is the value, $f(V)=\Sigma_{u\in V(G)}f(u)$. The {\em total Roman domination number} $\gamma_{tR}(G)$ of $G$ is the minimum weight of a total Roman dominating function of $G$. A subset $S$ of $V$ is a $2$-independent set of $G$ if every vertex of $S$ has at most one neighbor in $S$. The maximum cardinality of a $2$-independent set of $G$ is the $2$-independence number $\beta_2(G)$. These two parameters are incomparable in general, however, we show that if $T$ is a tree, then $\gamma_{tR}(T)\le \frac{3}{2}\beta_2(T)$ and we characterize all trees attaining the equality.The generous Roman domination number
https://toc.ui.ac.ir/article_27602.html
Let $G=(V,E)$\ be a simple graph and $f:V\rightarrow\{0,1,2,3\}$ be a function. A vertex $u$ with $f\left( u\right) =0$ is called an undefended vertex with respect to $f$ if it is not adjacent to a vertex $v$ with $f(v)\geq2.$ We call the function $f$ a generous Roman dominating function (GRDF) if for every vertex with $f\left( u\right) =0$ there exists at least a vertex $v$ with $f(v)\geq2$ adjacent to $u$ such that the function $f^{\prime}:V\rightarrow \{0,1,2,3\}$, defined by $f^{\prime}(u)=\alpha$, $f^{\prime}(v)=f(v)-\alpha$ where $\alpha=1$ or $2$, and $f^{\prime}(w)=f(w)$ if $w\in V-\{u,v\}$ has no undefended vertex. The weight of a generous Roman dominating function $f$ is the value $f(V)=\sum_{u\in V}f(u)$. The minimum weight of a generous Roman dominating function on a graph $G$\ is called the generous Roman domination number of $G$, denoted by $\gamma_{gR}\left( G\right) $. In this paper, we initiate the study of generous Roman domination and show its relationships. Also, we give the exact values for paths and cycles. Moreover, we present an upper bound on the generous Roman domination number, and we characterize cubic graphs $G$ of order $n$ with $\gamma_{gR}\left( G\right) =n-1$, and a Nordhaus-Gaddum type inequality for the parameter is also given. Finally, we study the complexity of this parameter.Note on skew-eigenvalues of digraphs
https://toc.ui.ac.ir/article_27603.html
Let $G^\sigma$ be an oriented graph with underlying simple graph $G$. The skew-adjacency matrix of $G^\sigma$ is the $\{0, 1, -1\}$-matrix $S=S(G^\sigma)=[s_{ij}]$, such that $s_{ij}=1$ if $(v_i, v_j)$ is an arc in $G^\sigma$, $s_{ij}=-1$ if $(v_j, v_i)$ is an arc in $G^\sigma$ and $s_{ij}=0$, otherwise. In this paper, all connected oriented graphs with three distinct skew-eigenvalues $0$ and $\pm 2 \mathbf{i}$ are characterized.Bijections for classes of labelled trees
https://toc.ui.ac.ir/article_27618.html
Trees are acyclic connected graphs. Plane trees, $d$-ary trees, binary trees, noncrossing trees and their generalizations, which are families of trees, have been enumerated by many authors using various statistics. These trees are known to be enumerated by Catalan or Catalan-like formulas (Fuss-Catalan numbers). One of the most common approaches to the enumeration of these trees is by means of generating functions. Another method that can be used to enumerate them is by constructing bijections between sets of the same cardinality. The bijective method is preferred to other methods by many combinatorialists. So, in this paper, we construct bijections relating $k$-plane trees, $k$-noncrossing increasing trees, $k$-noncrossing trees, $k$-binary trees and weakly labelled $k$-trees.The minimum $\varepsilon$-spectral radius of $t$-clique trees with given diameter
https://toc.ui.ac.ir/article_27700.html
The eccentricity matrix $\varepsilon(G)$ of a graph $G$ is defined as \begin{equation}\varepsilon(G)_{uv}= \begin{cases}d_{uv} &amp; d_{uv}=min\{e(u),e(v)\},\\0 &amp; d_{uv} &lt; min\{e(u),e(v)\}. \notag\end{cases}\end{equation} Let $T_t$ be a $t$-clique tree corresponding to the tree $T($underlying graph of $T_t)$ with order $n'=(n-1)t+1$ and diameter $d$. In this paper, we identify the extremal $t$-clique trees with given diameter having the minimum $\varepsilon$-spectral radius. Simultaneously, we calculate the lower bound of $\varepsilon$-spectral radius of $t$-clique trees when $n-d$ is odd.On the infinitary van der Waerden's Theorem
https://toc.ui.ac.ir/article_27705.html
We give a purely combinatorial proof for the infinitary van der Waerden's theorem.Peg solitaire on line graphs
https://toc.ui.ac.ir/article_27708.html
In $2011$, Beeler and Hoilman generalized the game of peg solitaire to arbitrary connected graphs. Since then peg solitaire and related games have been considered on many graph classes. One of the main goals is the characterization of solvable graphs. To this end, different graph operations, such as joins and Cartesian products, have been considered in the past. In this article, we continue this venue of research by investigating line graphs. Instead of playing peg solitaire on the line graph $L(G)$ of a graph $G$, we introduce a related game called stick solitaire and play it on $G$. This game is examined on several well-known graph classes, for example complete graphs and windmills. In particular, we prove that most of them are stick-solvable. We also present a family of graphs which contains unsolvable graphs in stick solitaire. Naturally, the Fool's stick solitaire number is an object of interest, which we compute for the previously considered graph classes.A closed formula for the number of inequivalent ordered integer quadrilaterals with fixed perimeter
https://toc.ui.ac.ir/article_27710.html
Given an integer $n\geq4$, how many inequivalent quadrilaterals with ordered integer sides and perimeter $n$ are there? Denoting such number by $Q(n)$, the answer is given by the following closed formula:\[Q(n)=\left\{ \dfrac{1}{576}n\left( n+3\right) \left( 2n+3\right) -\dfrac{\left( -1\right) ^{n}}{192}n\left( n-5\right) \right\} \cdot\]The inverse 1-median problem on a tree with transferring the weight of vertices
https://toc.ui.ac.ir/article_27767.html
In this paper, we investigate a case of the inverse 1-median problem on a tree by transferring the weights of vertices which has not been raised so far. This problem considers modifying the weights of vertices via transferring weights of the vertices with the minimum cost such that a given vertex of the tree becomes the 1-median with respect to the new weights. A linear programming model is proposed for this problem. The applicability and efficiency of the presented model are shown in numerical examples and a real-life problem dealing with transferring users in a social network.Methods for counting the intersections of slopes in the flat torus
https://toc.ui.ac.ir/article_27773.html
We define slopes in the flat torus as the set of equivalence classes of the solutions of linear equations in $\mathbb{R}^2$. The definition is equivalent to that of closed geodesics in the flat torus passing through the equivalence class of the point $(0,0)$. In this paper we derive formulas for counting the number of points in the intersection of multiple slopes in the flat torus.Approachable graph (tree) and It's application in hyper (network)
https://toc.ui.ac.ir/article_27872.html
A hypertree is a special type of connected hypergraph that removes&lrm; &lrm;any&lrm;, &lrm;its hyperedge then results in a disconnected hypergraph&lrm;. &lrm;Relation between hypertrees (hypergraphs) and trees (graphs) can be helpful to solve real problems in hypernetworks and networks and it is the main tool in this regard&lrm;. &lrm;The purpose of this paper is to introduce a positive relation (as $\alpha$-relation) on hypertrees that makes a connection between hypertrees and trees&lrm;. &lrm;This relation is dependent on some parameters such as path&lrm;, &lrm;length of a path&lrm;, &lrm;and the intersection of hyperedges&lrm;. &lrm;For any $q\in \mathbb{N}&lrm;, &lrm;$ we introduce the concepts of a derivable tree&lrm;, &lrm;$(\alpha&lrm;, &lrm;q)$-hypergraph&lrm;, &lrm;and fundamental $(\alpha&lrm;, &lrm;q)$-hypertree for the first time in this study and analyze the structures of derivable trees from hypertrees via given positive relation&lrm;. &lrm;In the final&lrm;, &lrm;we apply the notions of derivable trees&lrm;, &lrm;$(\alpha&lrm;, &lrm;q)$-trees in real optimization problems by modeling hypernetworks and networks based on hypertrees and trees&lrm;, &lrm;respectively.&lrm;&lrm;&lrm;Relations between energy of graphs and wener, harary indices
https://toc.ui.ac.ir/article_27873.html
Harary and Wiener indices are distance-based topological index. In this paper, we study the relations of graph energy $\varepsilon(G)$ and its Harary index $\textup{H}(G)$ and Wiener index $\textup{W}(G)$. Moreover, for a given graph $G$ we study the lower bound of $\frac{\textup{H}(G)}{\varepsilon(G)}$ and $\frac{\textup{W}(G)}{\varepsilon(G)}$ in terms of number of vertices of $G$.On the $sd_{b}$-critical graphs
https://toc.ui.ac.ir/article_27887.html
A $b$-coloring of a graph\ $G$ is a proper coloring of its vertices such that each color class contains a vertex that has a neighbor in every other color classes. The $b$-chromatic number of a graph $G$, denoted by $b(G)$, is the largest integer $k$ such that $G$ admits a $b$-coloring with $k$ colors. Let $G_{e}$ be the graph obtained from $G$ by subdividing the edge $e $. A graph $G$ is $sd_{b}$-critical if $b(G_{e})&lt;b(G)$ holds for any edge $e$ of $G$. In this paper, we first present \ several basic properties of $sd_{b} $-critical graphs and then we give a characterization of $sd_{b}$-critical $P_{4}$-sparse graphs and $sd_{b}$-critical quasi-line graphs.An existence theorem of perfect matching on $k$-partite $k$-uniform hypergraphs via distance spectral radius
https://toc.ui.ac.ir/article_27945.html
Let $n_1, n_2,\ldots,n_k$ be integers and $V_1, V_2,\ldots,V_k$ be disjoint vertex sets with $|V_i|=n_i$ for each $i= 1, 2,\ldots,k$. A $k$-partite $k$-uniform hypergraph on vertex classes $V_1, V_2,\ldots,V_k$ is defined to be the $k$-uniform hypergraph whose edge set consists of the $k$-element subsets $S$ of $V_1 \cup V_2 \cup \cdots \cup V_k$ such that $|S\cap V_i|=1$ for all $i= 1, 2,\ldots,k$. We say that it is balanced if $n_1=n_2=\cdots=n_k$. In this paper, we give a distance spectral radius condition to guarantee the existence of perfect matching in $k$-partite $k$-uniform hypergraphs, this result generalize the result of Zhang and Lin &nbsp;[Perfect matching and distance spectral radius in graphs and bipartite graphs, Discrete Appl. Math., 304 (2021) 315-322].