Given a connected graph $G$, a vertex $w\in V(G)$ strongly resolves two vertices $u,v\in V(G)$ if there exists some shortest $u-w$ path containing $v$ or some shortest $v-w$ path containing $u$. A set $S$ of vertices is a strong metric generator for $G$ if every pair of vertices of $G$ is strongly resolved by some vertex of $S$. The smallest cardinality of a strong metric generator for $G$ is called the strong metric dimension of $G$. In this paper we obtain several relationships between the strong metric dimension of the lexicographic product of graphs and the strong metric dimension of its factor graphs.
In a graph $G$, a vertex dominates itself and its neighbours. A subset $S\subseteq V(G)$ is said to be a double dominating set of $G$ if $S$ dominates every vertex of $G$ at least twice. The minimum cardinality among all double dominating sets of $G$ is the double domination number. In this article, we obtain tight bounds and closed formulas for the double domination number of lexicographic product graphs $G\circ H$ in terms of invariants of the factor graphs $G$ and $H$.
Given a simple and connected graph $G=(V,E)$, and a positive integer $k$, a set $S\subseteq V$ is said to be a $k$-metric generator for $G$, if for any pair of different vertices $u,v\in V$, there exist at least $k$ vertices $w_1,w_2,\ldots,w_k\in S$ such that $d_G(u,w_i)\ne d_G(v,w_i)$, for every $i\in \{1,\ldots,k\}$, where $d_G(x,y)$ denotes the distance between $x$ and $y$. The minimum cardinality of a $k$-metric generator is the $k$-metric dimension of $G$. A set $S\subseteq V$ is a $k$-adjacency generator for $G$ if any two different vertices $x,y\in V(G)$ satisfy $|((N_G(x)\triangledown…
A vertex $v$ of a graph $G=(V,E)$ is said to be undefended with respect to a function $f: V \longrightarrow \{0,1,2\}$ if $f(v)=0$ and $f(u)=0$ for every vertex $u$ adjacent to $v$. We call the function $f$ a weak Roman dominating function if for every $v$ such that $f(v)=0$ there exists a vertex $u$ adjacent to $v$ such that $f(u)\in \{1,2\}$ and the function $f': V \longrightarrow \{0,1,2\}$ defined by $f'(v)=1$, $f'(u)=f(u)-1$ and $f'(z)=f(z)$ for every $z\in V \setminus\{u,v\}$, has no undefended vertices. The weight of $f$ is $w(f)=\sum_{v\in V(G) }f(v)$. The weak Roman domination number …
The neighbourhood of a vertex $v$ of a graph $G$ is the set $N(v)$ of all vertices adjacent to $v$ in $G$. For $D\subseteq V(G)$ we define $\overline{D}=V(G)\setminus D$. A set $D\subseteq V(G)$ is called a super dominating set if for every vertex $u\in \overline{D}$, there exists $v\in D$ such that $N(v)\cap \overline{D}=\{u\}$. The super domination number of $G$ is the minimum cardinality among all super dominating sets in $G$. In this article we obtain closed formulas and tight bounds for the super dominating number of lexicographic product graphs in terms of invariants of the factor graphs…
We discuss how to find the well-covered dimension of a graph that is the Cartesian product of paths, cycles, complete graphs, and other simple graphs. Also, a bound for the well-covered dimension of $K_n\times G$ is found, provided that $G$ has a largest greedy independent decomposition of length $c<n$. Formulae to find the well-covered dimension of graphs obtained by vertex blowups on a known graph, and to the lexicographic product of two known graphs are also given.
A digraph $D$ is an efficient closed domination digraph if there exists a subset $S$ of $V(D)$ for which the closed out-neighborhoods centered in vertices of $S$ form a partition of $V(D)$. In this work we deal with efficient closed domination digraphs among several product of digraphs. We completely describe the efficient closed domination digraphs among lexicographic and strong products of digraphs. We characterize those direct products of digraphs that are efficient closed domination digraphs, where factors are either two cycles or two paths. Among Cartesian product of digraphs, we describe…
A vertex coloring of a graph $G$ is distinguishing if non-identity automorphisms do not preserve it. The distinguishing number, $D(G)$, is the minimum number of colors required for such a coloring and the distinguishing threshold, $θ(G)$, is the minimum number of colors~$k$ such that any arbitrary $k$-coloring is distinguishing. Moreover, $Φ_k (G)$ is the number of distinguishing coloring of $G$ using at most $k$ colors. In this paper, for some graph operations, namely, vertex-sum, rooted product, corona product and lexicographic product, we find formulae of the distinguishing number and thres…
A graph is closed when its vertices have a labeling by $[n]$ such that the binomial edge ideal $J_G$ has a quadratic Gröbner basis with respect to the lexicographic order induced by $x_1 > \cdots > x_n > y_1> \cdots > y_n$. In this paper, we generalize this notion and study the so called $m-$closed graphs. We find equivalent condition to $3-$closed property of an arbitrary tree $T$. Using it, we classify a class of $3-$closed trees. The primary decomposition of this class of graphs is also studied.
This paper is concerned with developing some new connection formulae between two generalized classes of Fibonacci and Lucas polynomials. All the connection coefficients involve hypergeometric functions of the type $_2F_{1}(z)$, for certain $z$. Several new connection formulae between some famous polynomials such as Fibonacci, Lucas, Pell, Fermat, Pell-Lucas, and Fermat-Lucas polynomials are deduced as special cases of the derived connection formulae. Some of the introduced formulae generalize some of those existing in the literature. As two applications of the derived connection formulae, some…
Let $G=(V,E)$ be a graph of order $n$. A distance magic labeling of $G$ is a bijection $\ell \colon V\rightarrow {1,...,n}$ for which there exists a positive integer $k$ such that $\sum_{x\in N(v)}\ell (x)=k$ for all $v\in V $, where $N(v)$ is the neighborhood of $v$. We introduce a natural subclass of distance magic graphs. For this class we show that it is closed for the direct product with regular graphs and closed as a second factor for lexicographic product with regular graphs. In addition, we characterize distance magic graphs among direct product of two cycles.
We consider a scalar-valued implicit function of many variables, and provide two closed formulae for all of its partial derivatives. One formula is based on products of partial derivatives of the defining function, the other one involves less products of building blocks of multinomial type, and we study the combinatorics of the coefficients showing up in both formulae.
The General Randić index $R_α$ of a simple graph $G$ is defined as \[ R_α(G)=\sum_{v_{i}\sim v_{j}} (δ_{i}δ_{j})^α, \] where $δ_i$ denotes the degree of the vertex $v_i$. Rodríguez-Velázquez and Tomás-Andreu [MATCH Commun. Math. Comput. Chem. 74 (1) (2015) 145--160] obtained closed formulae for the Randić index $R_{-1/2}$ of Sierpiński-type polymeric networks, where the base graph is a complete graph, a triangle-free regular graph or a bipartite semiregular graph. In the present article we obtain closed formulae for the general Randić index $R_α$ of Sierpiński-type polymeric networks, where th…
In this paper we obtain closed formulae for several parameters of generalized Sierpiński graphs $S(G,t)$ in terms of parameters of the base graph $G$. In particular, we focus on the chromatic, vertex cover, clique and domination numbers.
The floor and ceiling functions appear often in mathematics and manipulating sums involving floors and ceilings is a subtle game. Fortunately, the well-known textbook Concrete Mathematics provides a nice introduction with a number of techniques explained and a number of single or double sums treated as exercises. For two such double sums we provide their single-sum analogues. These closed-form identities are given in terms of a dual partition of the multiset (regarded as a partition) of all b-ary digits of a nonnegative integer. We also present the double- and single-sum analogues involving th…
Asymptotics for graphically divergent series: dense digraphs and 2-SAT formulae
LaBRI, Université de Bordeaux（波尔多大学）；LIPN, Université Sorbonne Paris Nord（索邦巴黎北大学）；IMB, Université de Bourgogne（勃艮第大学）；LIB, Université de Bourgogne（勃艮第大学）；LIP6, Sorbonne Université（索邦大学）
A generator of a metric space is a set $S$ of points in the space with the property that every point of the space is uniquely determined by its distances from the elements of $S$. Given a simple graph $G=(V,E)$, we define the distance function $d_{G,2}:V\times V\rightarrow \mathbb{N}\cup \{0\}$, as $d_{G,2}(x,y)=\min\{d_G(x,y),2\},$ where $d_G(x,y)$ is the length of a shortest path between $x$ and $y$ and $\mathbb{N}$ is the set of positive integers. Then $(V,d_{G,2 })$ is a metric space. We say that a set $S\subseteq V$ is a $k$-adjacency generator for $G$ if for every two vertices $x,y\in V$…
In this article, we obtain closed formulae for the Italian domination number of rooted product graphs. As a particular case of the study, we derive the corresponding formulas for corona graphs, and we provide an alternative proof that the problem of computing the Italian domination number of a graph is NP-hard.
A circulant is a Cayley graph over a cyclic group. A well-covered graph is a graph in which all maximal stable sets are of the same size, or in other words, they are all maximum. A CIS graph is a graph in which every maximal stable set and every maximal clique intersect. It is not difficult to show that a circulant G is a CIS graph if and only if G and its complement are both well-covered and the product of the independence and the clique numbers of G is equal to the number of vertices. It is also easy to demonstrate that both families, the circulants and the CIS graphs, are closed with respec…
