In this paper, we obtain the following upper bounds for the largest Laplacian graph eigenvalue: $$\mu \le \underset{i}{max}\left\{\sqrt{2d_i(m_i+d_i)+n–2d_i–2\sum _{j:j\sim i}|N_i\cap N_j|}\right\}$$ where $d_i$ and $m_i$ are the degree of vertex $i$ and the average degree of vertex $i$, respectively; $|N_i\cap N_j|$ is the number of common neighbors of vertices $i$ and $j$. We also compare this bound with some known upper bounds.

A three-colored digraph $D$ is primitive if and only if there exist nonnegative integers $h$, $k$, and $v$ with $h+k+v>0$ such that for each pair $(i,j)$ of vertices there is an $(h,k,v)$-walk in $D$ from $i$ to $j$. The exponent of the primitive three-colored digraph $D$ is defined to be the smallest value of $h+k+v$ over all such $h$, $k$, and $v$. In this paper, a class of special primitive three-colored digraphs with $n$ vertices, consisting of one $n$-cycle and two $(n-1)$-cycles, are considered. For the case $a=c–1$, some primitive conditions, the tight upper bound on the exponents, and the characterization of extremal three-colored digraphs are given.

Skew-quasi-cyclic codes over a finite field are viewed as skew-cyclic codes on a noncommutative ring of matrices over a finite field. This point of view gives a new construction of skew-quasi-cyclic codes. Let ${\mathbb{F}}_q$ be the Galois field with $q$ elements and $\theta$ be an automorphism of ${\mathbb{F}}_q$. We propose an approach to consider the relationship between left ideals in $M_l({\mathbb{F}}_q)[X,\theta ]/(X^s–1)$ and skew-quasi-cyclic codes of length $ls$ and index $l$ over ${\mathbb{F}}_q$, under $\theta$, which we denote by $\theta$-SQC codes (or SQC codes for short when there is no ambiguity). We introduce the construction of SQC codes from the reversible divisors of $X^s–1$ in $M_l({\mathbb{F}}_q)[X,\theta ]$. In addition, we give an algorithm to search for the generator polynomials of general SQC codes.

In this paper, we investigate the concepts of $k$-limited packing and $k$-tuple domination in graphs and give several bounds on the size of them. These bounds involve many well-known parameters of graphs. Also, we establish a connection between these concepts that implies some new results in this area. Finally, we improve many bounds in the literature.

Let $G=(V,E)$ be a simple graph. A paired-dominating set of a graph $G$ is a dominating set whose induced subgraph contains a perfect matching. The paired domination number of a graph $G$, denoted by ${\gamma }_p(G)$, is the minimum cardinality of a paired-dominating set in $G$. In this paper, we study the paired domination number of generalized Petersen graphs $P(n,2)$ and prove that for any integer $n\ge 6$, ${\gamma }_p(P(n,2))=2\lfloor \frac{n}{3}\rfloor +n(\mathrm{mod}3)$.

The Estrada index of a simple connected graph $G$ of order $n$ is defined as $EE(G)=\sum _{i=1}^n{e}^{{\lambda }_i}$, where ${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n$ are the eigenvalues of the adjacency matrix of $G$. In this paper, we characterize all pentacyclic graphs of order $n$ with maximal Estrada index.

Let $\mathrm{\Pi }$ be a finite polar space of rank $n\ge 2$ fully embedded into a projective space $\mathrm{\Sigma }$. In this note, we determine all tight sets of $\mathrm{\Pi }$ of the form $({\mathrm{\Sigma }}_1\cap \mathcal{P})\cup ({\mathrm{\Sigma }}_2\cap \mathcal{P})$, where $\mathcal{P}$ denotes the point set of $\mathrm{\Pi }$ and ${\mathrm{\Sigma }}_1,{\mathrm{\Sigma }}_2$ are two mutually disjoint subspaces of $\mathrm{\Sigma }$. In this way, we find two families of $2$-tight sets of elliptic polar spaces that were not described before in the literature.

In this paper, we define a new matrix identity for bi-periodic Fibonacci and Lucas numbers. By using the matrix method, we give simple proofs of several properties of these numbers. Moreover, we obtain a new binomial sum formula for bi-periodic Fibonacci and Lucas numbers, which generalize the former results.

Hein and Sarvate show how to decompose $\lambda$ copies of a complete graph $K_n$, for some minimal value of $\lambda$, into so-called LOE and OLE graphs. In this paper, we will show that for all possible values of $\lambda$, the necessary conditions are sufficient for the LOE and OLE decompositions.

Let $R$ be a commutative ring. The regular digraph of ideals of $R$, denoted by $\mathcal{R}(R)$, is a digraph whose vertex-set is the set of all non-trivial ideals of $R$ and, for every two distinct vertices $I$ and $J$, there is an arc from $I$ to $J$, whenever $I$ contains a non-zero divisor of $J$. In this paper, we investigate the planarity of $\mathcal{R}(R)$. We also completely characterize the rings $R$ such that $\mathcal{R}(R)$ is a ring graph, and the situations under which the genus of $\mathcal{R}(R)$ is finite. Moreover, we study the independence number and the girth of $\mathcal{R}(R)$, and also we find all cases that $\mathcal{R}(R)$ is bipartite.

In this paper, the existence of Yang Hui type magic squares of order $n$ with $t$-powered sum (YMS($n$, $t$)) for general $t$ is investigated. Some constructions of YMS($n$, $t$) are obtained by using strongly symmetric self-orthogonal diagonal Latin squares and magic rectangles. Applying these constructions, it is proved that for an integer $t>1$ there exist both a symmetric elementary YMS($2^t$, $2t–2$) and a symmetric elementary YMS($2^t–k$, $2t$) for odd $k>1$, which improves the known result on YMSs.

To gain a better understanding of clean rings and their relatives, the clean graph of a commutative ring with identity is introduced and its various properties are established. Further investigation of clean graphs leads to additional results concerning other classes of rings.

For a connected graph, the distance spectral radius is the largest eigenvalue of its distance matrix. In this paper, of all trees with both given order and fixed diameter, the trees with the minimal distance spectral radius are completely characterized.

In this paper, a domination-type parameter, called dynamical $2$-domination number, will be introduced. Let $G=(V(G),E(G))$ be a graph. A subset $D\subseteq V(G)$ is called a $2$-dominating set in $G$ if every vertex in $V(G)\setminus D$ is adjacent to at least two vertices in $D$, and in this paper $D$ is called a dynamical $2$-dominating set if there exists a sequence of sets $D=V_0\subseteq V_1\subseteq V_2\subseteq \cdots \subseteq V_k=V(G)$ such that, for each $i$, $V_{i-1}$ is a $2$-dominating set in $⟨V_i⟩$, the induced subgraph generated by $V_i$. Also, for a given graph $G$, the size of its dynamical $2$-dominating sets of minimum cardinality will be called the dynamical $2$-domination number of $G$ and will be denoted by ${\overline{\gamma }}_2(G)$. We study some basic properties of dynamical $2$-dominating sets and compute ${\overline{\gamma }}_2(G)$ for some graph classes. Also, some results about ${\overline{\gamma }}_2$ of a number of binary operations on graphs are proved. A characterization of graphs with extreme values of ${\overline{\gamma }}_2$ is presented. Finally, we study this concept for trees and give an upper bound and a lower bound for the dynamical $2$-domination number of trees.

A graph $G$ is said to be equitably $k$-colorable if the vertex set of $G$ can be divided into $k$ independent sets for which any two sets differ in size at most one. The equitable chromatic number of $G$, ${\chi }_{=}(G)$, is the minimum $k$ for which $G$ is equitably $k$-colorable. The equitable chromatic threshold of $G$, ${\chi }_m^{\ast }(G)$, is the minimum $k$ for which $G$ is equitably $k^{\prime }$-colorable for all $k^{\prime }\ge k$. In this paper, the exact values of ${\chi }_m^{\ast }(P_{n^{\prime },2}\mathrm{<img\; draggable="false"\; role="img"\; class="emoji"\; alt="◻"\; src="https://s.w.org/images/core/emoji/15.0.3/svg/25fb.svg">}{K}_{m,n})$ and ${\chi }_{=}(P_{n^{\prime },m}\mathrm{<img\; draggable="false"\; role="img"\; class="emoji"\; alt="◻"\; src="https://s.w.org/images/core/emoji/15.0.3/svg/25fb.svg">}{K}_{m,n})$ are obtained except that $3\le {\xi }_m^{\ast }(P_{5,2}\mathrm{<img\; draggable="false"\; role="img"\; class="emoji"\; alt="◻"\; src="https://s.w.org/images/core/emoji/15.0.3/svg/25fb.svg">}{K}_{m,n})={\chi }_{=}(P_{s,m}\mathrm{<img\; draggable="false"\; role="img"\; class="emoji"\; alt="◻"\; src="https://s.w.org/images/core/emoji/15.0.3/svg/25fb.svg">}{K}_{m,n})\le 4$ when $m+n\ge 3min\{m,n\}+2$ or $m+n<3min\{m,n\}–2$.

The sum-connectivity energy of a graph is defined as the sum of the absolute value of all the eigenvalues of its sum-connectivity matrix. In this paper, we give further lower and upper bounds for the sum-connectivity energy in terms of the number of vertices, number of edges, the harmonic index, and determinant of the sum-connectivity matrix. We also show that among connected graphs with $n$ vertices, the star graph $K_{1,n-1}$ has the minimum sum-connectivity energy.

A graph is one-regular if its automorphism group acts regularly on the set of its arcs. In this paper, $4$-valent one-regular graphs of order $5p^2$, where $p$ is a prime, are classified.

In this paper, we obtain that the characteristic polynomials of the signless Laplacian matrix of $Q(G)$, $R(G)$, $T(G)$ can be expressed in terms of the characteristic polynomial of $G$ when $G$ is a regular or semiregular graph, from which upper bounds for the incidence energy of $Q(G)$, $R(G)$, $T(G)$ are deduced.

Let $R$ be a commutative ring with unity. The co-maximal ideal graph of $R$, denoted by $\mathrm{\Gamma }(R)$, is a graph whose vertices are the proper ideals of $R$ which are not contained in the Jacobson radical of $R$, and two vertices $I_1$ and $I_2$ are adjacent if and only if $I_1+I_2=R$. We classify all commutative rings whose co-maximal ideal graphs are planar. In 2012, the following question was posed: If $\mathrm{\Gamma }(R)$ is an infinite star graph, can $R$ be isomorphic to the direct product of a field and a local ring? In this paper, we give an affirmative answer to this question.

In this paper, a generalization of the Stirling numbers of the first and second kind, called $m$ -Stirling numbers of the first and second kind, are derived. Based on the colored base-$m$ number system, we give a combinatorial interpretation of $m$ -Stirling numbers of the second kind. Some basic properties of the two kinds of $m$ -Stirling numbers, including generating functions, explicit expressions, and recurrence relations, are also obtained.

A proper $k$-total coloring of a simple graph $G$ is called $k$-vertex-distinguishing proper total coloring ($k$-VDTC) if for any two distinct vertices $u$ and $v$ of $G$, the set of colors assigned to $u$ and its incident edges differs from the set of colors assigned to $v$ and its incident edges. The minimum number of colors required for a vertex-distinguishing proper total coloring of $G$, denoted by ${\chi }_{vt}(G)$, is called the vertex-distinguishing proper total chromatic number. For $p$ even, $p\ge 4$ and $q\ge 3$, we will obtain vertex-distinguishing proper total chromatic numbers of complete $p$-partite graphs with each part of cardinality $q$.

Let $G$ be a simple graph of order $n$. The domination polynomial of $G$ is the polynomial $D(G,x)=\sum _{i=0}^{n}d(G,i){\lambda }^i$, where $d(G,i)$ is the number of dominating sets of $G$ of size $i$. Every root of $D(G,\lambda )$ is called a domination root of $G$. It is clear that $(0,\mathrm{\infty })$ is a zero-free interval for the domination polynomial of a graph. It is interesting to investigate graphs that have complex domination roots with positive real parts. In this paper, we first investigate the complexity of the domination polynomial at specific points. Then, we present and investigate some families of graphs whose complex domination roots have positive real parts.

A quasi-tree is a graph for which the deletion of some vertex results in a tree. We determine the unique graph with minimum distance spectral radius among quasi-trees with fixed order and the unique graph with maximum distance spectral radius among cycle-containing quasi-trees with fixed order.

We initiate the study of double outer-independent domination in graphs. A vertex of a graph is said to dominate itself and all of its neighbors. A double outer-independent dominating set of a graph $G$ is a set $D$ of vertices of $G$ such that every vertex of $G$ is dominated by at least two vertices of $D$, and the set $V(G)\setminus D$ is independent. The double outer-independent domination number of a graph $G$ is the minimum cardinality of a double outer-independent dominating set of $G$. First, we discuss the basic properties of double outer-independent domination in graphs. We find the double outer-independent domination numbers for several classes of graphs. Next, we prove lower and upper bounds on the double outer-independent domination number of a graph, and we characterize the extremal graphs. Then, we study the influence of removing or adding vertices and edges. We also give Nordhaus-Gaddum type inequalities.

For any two graphs $F_1$ and $F_2$, the graph Ramsey number $r(F_1,F_2)$ is the smallest positive integer $N$ with the property that every graph of at least $N$ vertices contains $F_1$ or its complement contains $F_2$ as a subgraph. In this paper, we consider the Ramsey numbers for theta-complete graphs. In fact, we prove that $r({\theta }_n,K_5)=4n-3$ for $n\ge 6$ and $n\ge 10$.

The Randić index $R$ is an important topological index in chemistry. In order to attack some conjectures concerning the Randić index, a modification $R^{\prime }$ of this index was introduced by Dvorak et al. [6]. The $R^{\prime }$ index of a graph $G$ is defined as the sum of the weights $\frac{1}{max\{d(u)d(v)\}}$ of all edges $uv$ of $G$, where $d(u)$ denotes the degree of a vertex $u$ in $G$. We first give a best possible lower bound of $R^{\prime }$ for a graph with minimum degree at least two and characterize the corresponding extremal graphs, and then we establish some relations between $R^{\prime }$ and the chromatic number, the girth of a graph.

There are operations that transform a map $\mathcal{M}$ (an embedding of a graph on a surface) into another map on the same surface, modifying its structure and consequently its set of flags $\mathcal{F}\mathcal{(}\mathcal{M}\mathcal{)}$. For instance, by truncating all the vertices of a map $\mathcal{M}$, each flag in $\mathcal{F}\mathcal{(}\mathcal{M}\mathcal{)}$ is divided into three flags of the truncated map. Orbanić, Pellicer, and Weiss studied the truncation of $k$-orbit maps for $k\le 3$. They introduced the notion of $T$-compatible maps in order to give a necessary condition for a truncation of a $k$-orbit map to be either $k$-, $\frac{3k}{2}$-, or $3k$-orbit map. Using a similar notion, by introducing an appropriate partition on the set of flags of the maps, we extend the results on truncation of $k$-orbit maps for $k\le 7$ and $k=9$.

Let ${\mathrm{\Re }}_{\beta }$ denote the set of trees on $n=kG+1$ ($k\ge 2$) vertices with matching number $\beta$. In this paper, the trees with minimal spectral radius among ${\mathrm{\Re }}_{\beta }$ ($2\le \delta \le 4$) are determined, respectively.

Generalized whist tournament designs and ordered whist tournament designs are relatively new specializations of whist tournament designs, having first appeared in $2003$ and $1996$, respectively. In this paper, we extend the concept of an ordered whist tournament to a generalized whist tournament and introduce an entirely new combinatorial design, which we call a generalized ordered whist tournament. We focus specifically on generalized whist tournaments for games of size $6$ and teams of size $3$, where the number of players is a prime of the form $6n+1$, and prove that these tournaments exist for all primes $p$ of the form $p=6n+1$, with the possible exception of $p\in \{7,13,19,37,61,67\}$.

A regular graph $\mathrm{\Gamma }$ is said to be semisymmetric if its full automorphism group acts transitively on its edge set but not on its vertex set. Some authors classified semisymmetric cubic graphs of orders $10p$ and $10p^2$. Also, it is proved that there is no connected semisymmetric cubic graph of order $10p^3$. In this paper, we continue this work and prove that there is no connected semisymmetric cubic graph of order $10p^n$, where $n\ge 4$, $p\ge 7$, and $p\ne 11$.

In this.paper, by joint tree model, we obtain the genera of two types of graphs, which are suspensions of cartesian products of two types of bipartite graphs from a vertex.

Let $G$ be a connected graph with a perfect matching on $2n$ vertices ($n\ge 2$). A graph $G^{\prime }$ is a contraction of $G$ if it can be obtained from $G$ by a sequence of edge contractions. Then $G$ is said to be edge contractible if for any contraction $G^{\prime }$ of $G$ with $|V(G^{\prime })|$ even, $G^{\prime }$ has a perfect matching. In this note, we obtain a sufficient and necessary condition for a graph to be an edge contractible graph.

All finite Jacobson graphs with a Hamiltonian cycle or path, or Eulerian tour or trail are determined, and it is shown that a finite Jacobson graph is Hamiltonian if and only if it is pancyclic. Also, the length of the longest induced cycles and paths in finite Jacobson graphs are obtained.

A vertex subset $S$ of a digraph $D$ is called a dominating set of $D$ if every vertex not in $S$ is adjacent from at least one vertex in $S$. The domination number of $D$, denoted by $\gamma (D)$, is the minimum cardinality of a dominating set of $D$. We characterize the rooted trees and connected contrafunctional digraphs $D$ of order $n$ satisfying $\gamma (D)=\lceil \frac{n}{2}\rceil$. Moreover, we show that for every digraph $D$ of order $n$ with minimum in-degree at least one, $\gamma (D)\le \frac{(k+1)n}{2k+1}$, where $2k+1$ is the length of a shortest odd directed cycle in $D$, and we characterize the corresponding digraphs achieving this upper bound. In particular, if $D$ contains no odd directed cycles, then $\gamma (D)\le \frac{n}{2}$.

A graph is called degree-magic if it admits a labelling of the edges by integers $\{1,2,\dots ,|E(G)|\}$ such that the sum of the labels of the edges incident with any vertex $v$ is equal to $(1+|E(G)|)/2\mathrm{deg}(v)$. In this paper, we show that a class of join graphs are degree-magic.

A vertex-deleted unlabeled subgraph of a graph $G$ is called a card of $G$. A card of $G$ with which the degree of the deleted vertex is also given is called a degree-associated card or dacard of $G$. The degree-associated reconstruction number, $\mathrm{d}\mathrm{r}\mathrm{n}(G)$, of a graph $G$ is the size of the smallest collection of dacards of $G$ that uniquely determines $G$. The maximal subgraph without end vertices of a graph $G$ that is not a tree is called the pruned graph of $G$. It is shown that $\mathrm{d}\mathrm{r}\mathrm{n}$ of some connected graphs with regular pruned graph is $2$ or $3$.

The Wiener index of a connected graph is the sum of distances between all pairs of vertices in the graph. Feng et al. in [The hyper-Wiener index of bicyclic graphs, Utilitas Math., $84(2011)97-104$] determined the bicyclic graphs having the largest Wiener index. In this article, we determine the graphs having the second up to seventh largest Wiener indices among all bicyclic graphs with $n$ vertices.

The matching energy of a graph was introduced by Gutman and Wagner in $2012$ and defined as the sum of the absolute values of zeros of its matching polynomial. In this paper, we completely determine the graph with minimum matching energy in tricyclic graphs with given girth and without $K_4$-subdivision.

In this paper, we define and study the Gaussian Fibonacci and Gaussian Lucas $p$-numbers. We give generating functions, Binet formulas, explicit formulas, matrix representations, and sums of Gaussian Fibonacci $p$-numbers by matrix methods. For $p=1$, these Gaussian Fibonacci and Gaussian Lucas $p$-numbers reduce to the Gaussian Fibonacci and the Gaussian Lucas numbers.

Let $G$ be a graph of order $n$ and let $Q(G,x)=det(xI–Q(G))=\sum _{i=0}^n(-1{)}^i{\zeta }_i(G)x^{n-i}$ be the characteristic polynomial of the signless Laplacian matrix of $G$. We show that the Lollipop graph, $L_{n,3}$, has the maximal $Q$-coefficients, among all unicyclic graphs of order $n$ except $C_n$. Moreover, we determine graphs with minimal $Q$-coefficients, among all unicyclic graphs of order $n$.

Let $G$ be a graph with $n$ vertices, $\mathcal{G}(G)$ the subdivision graph of $G$. $V(G)$ denotes the set of original vertices of $G$. The generalized subdivision corona vertex graph of $G$ and $H_1,H_2,\dots ,H_n$ is the graph obtained from $\mathcal{G}(G)$ and $H_1,H_2,\dots ,H_n$ by joining the $i$th vertex of $V(G)$ to every vertex of $H_i$. In this paper, we determine the Laplacian (respectively, the signless Laplacian) characteristic polynomial of the generalized subdivision corona vertex graph. As an application, we construct infinitely many pairs of cospectral graphs.

In the paper, we show that the orientable genus of the generalized Petersen graph $P(km,m)$ is at least $\frac{km}{4}–\frac{m}{2}-\frac{km}{4m-4}+1$ if $m\ge 4$ and $k\ge 3$. We determine the orientable genera of $P(3m,m)$, $P(4k,4)$, $P(4m,m)$ if $m\ge 4$, $P(6m,m)$ if $m\equiv 0(\mathrm{mod}2)$ and $m\ge 6$, and so on.

Assume that ${\mu }_1,{\mu }_2,\dots ,{\mu }_n$ are the eigenvalues of the Laplacian matrix of a graph $G$. The Laplacian Estrada index of $G$, denoted by $LEE(G)$, is defined as $LEE(G)=\sum _{i=1}^n{e}^{{\mu }_i}$. In this note, we give an upper bound on $LEE(G)$ in terms of chromatic number and characterize the corresponding extremal graph.

In this note, we provide a combinatorial proof of a recent formula for the total number of peaks and valleys (either strict or weak) within the set of all compositions of a positive integer into a fixed number of parts.

The adjacent vertex distinguishing total chromatic number ${\chi }_{at}(G)$ of a graph $G$ is the smallest integer $k$ for which $G$ admits a proper $k$-total coloring such that no pair of adjacent vertices are incident to the same set of colors. Snarks are connected bridgeless cubic graphs with chromatic index $4$. In this paper, we show that ${\chi }_{at}(G)=5$ for two infinite subfamilies of snarks, i.e., the Loupekhine snark and Blanusa snark of first and second kind. In addition, we give an adjacent vertex distinguishing total coloring using $5$ colors for Watkins snark and Szekeres snark, respectively.

Let $G$ be a tricyclic graph. Tricyclic graphs are connected graphs in which the number of edges equals the number of vertices plus two. In this paper, we determine graphs with the largest signless Laplacian spectral radius among all the tricyclic graphs with $n$ vertices and diameter $d$.

A pebbling move on a graph $G$ consists of taking two pebbles off one vertex and placing one on an adjacent vertex. The pebbling number of a graph $G$, denoted by $f(G)$, is the least integer $n$ such that, however $n$ pebbles are located on the vertices of $G$, we can move one pebble to any vertex by a sequence of pebbling moves. For any connected graphs $G$ and $H$, Graham conjectured that $f(G\times H)\le f(G)f(H)$. In this paper, we give the pebbling number of some graphs and prove that Graham’s conjecture holds for the middle graphs of some even cycles.

Graph embedding is an important factor to evaluate the quality of an interconnection network. It is also a powerful tool for implementation of parallel algorithms and simulation of different interconnection networks. In this paper, we compute the exact wirelength of embedding circulant networks into cycle-of-ladders.

In this paper, we characterize the extremal digraph with the maximal signless Laplacian spectral radius and the minimal distance signless Laplacian spectral radius among all simple connected digraphs with a given dichromatic number, respectively.

Given a graph $G=(V,E)$ with no isolated vertex, a subset $S\subseteq V$ is a total dominating set of $G$ if every vertex in $V$ is adjacent to a vertex in $S$. A total dominating set $S$ of $G$ is a locating-total dominating set if for every pair of distinct vertices $u$ and $v$ in $V–S$, we have $N(u)\cap S\ne N(v)\cap S$, and $S$ is a differentiating-total dominating set if for every pair of distinct vertices $u$ and $v$ in $V$, we have $N(u)\cap S\ne N(v)\cap S$. The locating-total domination number (or the differentiating-total domination number) of $G$, denoted by ${\gamma }_t^L(G)$ (or ${\gamma }_t^D(G)$), is the minimum cardinality of a locating-total dominating set (or a differentiating-total dominating set) of $G$. In this paper, we investigate the bounds of locating and differentiating-total domination numbers of unicyclic graphs.