β-Differential of a Graph

Let k be a positive integer, and let G be a simple graph with vertex set V (G). A Roman k-dominating function on G is a function f : V (G) <TEX>$\rightarrow$</TEX> {0, 1, 2} such that every vertex u for which f(u) = 0 is adjacent to at least k vertices <TEX>$\upsilon_1,\;\upsilon_2,\;{\ldots},\;\upsilon_k$</TEX> with <TEX>$f(\upsilon_i)$</TEX> = 2 for i = 1, 2, <TEX>$\ldot$</TEX>, k. The weight of a Roman k-dominating function is the value f(V (G)) = <TEX>$\sum_{u{\in}v(G)}$</TEX> f(u). The minimum weight of a Roman k-dominating function on a graph G is called the Roman k-domination number <TEX>${\gamma}_{kR}$</TEX>(G) of G. Note that the Roman 1-domination number <TEX>$\gamma_{1R}$</TEX>(G) is the usual Roman domination number <TEX>$\gamma_R$</TEX>(G). In this paper, we investigate the properties of the Roman k-domination number. Some of our results extend these one given by Cockayne, Dreyer Jr., S. M. Hedetniemi, and S. T. Hedetniemi [2] in 2004 for the Roman domination number.

Let G be a simple graph with vertex set V(G) and edge set E(G). A vertex coloring of G is called a star coloring of G if any of the paths of 4 order are bicolored. The minimum number of colors required for a star coloring of G is denoted by χs (G). The corona product of simple graphs G of order m and H of order n is graph G ∘ H with vertex set V(G ∘ H) = {vi |i = 1,2,⋯m}∪{vij |i = 1,2,⋯m, j = 1,2,⋯n}, in which vi is adjacent to every vertex of Hi if and only if, vi ∈ V(G), vij ∈ V(Hi ). According to the existing graph dyeing literature, it has become a very important technical means to study the graph dyeing problem by using the graph structure operation. Therefore, it is of great significance to study the star coloring of graphs for studying the acyclic coloring and distance coloring of graphs, the study has strong application background and great theoretical value for computing graphs. In this paper, we find the upper bound of χs (G ∘ H) and the exact values of χs (G ∘ H) of the corona product G ∘ H of two graphs G and H as: χs (G ∘ H) ≤ χs (G) + χs (H); χs (Pm ∘ H) = χs (H) + 2; χs (K 1,m ∘ H) = χs (H) + 2; χs (Cn ∘ H) = χs (H) + 2, where n ≠ 5.

Let $G=(V(G),E(G))$ be a finite simple undirected graph with vertex set $V(G)$, edge set $E(G)$ and vertex subset $S\subseteq V(G)$. $S$ is termed \emph{open-dominating} if every vertex of $G$ has at least one neighbor in $S$, and \emph{open-independent, open-locating-dominating} (an $OLD_{oind}$-set for short) if no two vertices in $G$ have the same set of neighbors in $S$, and each vertex in $S$ is open-dominated exactly once by $S$. The problem of deciding whether or not $G$ has an $OLD_{oind}$-set has important applications that have been reported elsewhere. As the problem is known to be $\mathcal{NP}$-complete, it appears to be notoriously difficult as we show that its complexity remains the same even for just planar bipartite graphs of maximum degree five and girth six, and also for planar subcubic graphs of girth nine. Also, we present characterizations of both $P_4$-tidy graphs and the complementary prisms of cographs that have an $OLD_{oind}$-set.

Minimum face-spanning subgraphs of plane graphs

Boolean Graph Operator B

Let G be a simple graph with vertex set V(G) and edge set E(G). Automorphism of G is an isomorphism from G to itself. In other words, an automorphism of G is a permutation φ of V(G), which has the property that uv ∈ E(G) if and only if φ(u)φ(v) ∈ E(G), for every u, v ∈ V(G) that is φ preserves adjacency. The complete n-ary tree is a rooted-tree where its every internal vertex has exactly n children. The height of a rooted tree is the number of edges on longest path connecting the root to a leaf (a vertex of degree one). In this paper, we determine the number of automorphism of complete 2,3,4-ary trees with height h, for any positive integer h. We also conjecture the general formula to determine the number of automorphism of complete n-ary trees, for any n ≥ 5.

ABSTRACTLet be a simple graph with vertex set and edge set . By orienting a subset of , we get a mixed graph . Let be the Hermitian-adjacency matrix of , the rank of , written as , is called the H-rank of . Denote by the dimension of cycle spaces of G, that is , where denotes the number of connected components of G. Let be the independence number of G. In this paper, the relationship between the H-rank of and the independence number of G is characterized. Firstly, it is proved that for every mixed graph . Secondly, all the mixed graphs which satisfy are characterized.

Let G be a simple graph with vertex set V(G) and edge set E(G). Let ⟨ℤ2, +,*⟩ be a field with two elements. A vertex labeling f : V(G) → ℤ2 induces two edge labelings f+: E(G) → ℤ2 such that f+ (xy) = f(x) + f(y), whereas f* : E(G) → ℤ2 such that f* (xy) = f(x) f(y), for each edge xy ∈ E(G). For i ϵ ℤ2, let and . A labeling f of a graph G is said to be friendly if |υf (0) −υf (1)|≤ 1. The friendly index set of the graph G, denoted FI(G), is defined as the vertex labeling f is friendly}. This is a generalization of graph cordiality. The corresponding multiplicative version is called the product-cordial index set, denoted PCI(G), defined as the vertex labeling f is friendly}. In this paper, we investigate the friendly index and product-cordial index sets of a family of cubic graphs known as Möbius-liked graph, MG(n) for even n ≥ 4.

Let [Formula: see text] be a simple graph with vertex set [Formula: see text] and edge set [Formula: see text]. An outer-paired dominating set [Formula: see text] of a graph [Formula: see text] is a dominating set such that the subgraph induced by [Formula: see text] has a perfect matching. The outer-paired domination number of [Formula: see text], denoted by [Formula: see text], is the minimum cardinality of an outer-paired dominating set of [Formula: see text]. In this paper, we study the outer-paired domination number of graphs and present some sharp bounds concerning the invariant. Also, we characterize all the trees with [Formula: see text].

Let G = (V, E) be a connected and simple graphs with vertex set V and edge set E. A coloring of graph G is rainbow connected if there is a rainbow path that connects each two vertices of graph G. The minimum k such that G has a rainbow-connected using k colors of the edges of G is the rainbow connection number rc(G) of G. A graph with a bijective mapping f : E → {1, 2, …, |E|}. The sums of each paired vertex has distinct value, defined as ∑ e ∈E(v)f(e). Thus, the function of G clearly an antimagic labeling if the sums of each paired vertex has distinct value. It is clear that rainbow antimagic connection number is the smallest number of colors which are needed to make G rainbow connected, denoted by rcA (G). A bijection function f : E → {1, 2, …, |E|} is called a rainbow antimagic labeling if there is a rainbow path between every pair of vertices and for each edge e = uv ∈ E(G), the weight w(e) = f(u) + f(v). A graph G is rainbow antimagic if G has a rainbow antimagic labeling. In this paper, we will analyze the rainbow antimagic coloring of related book graph.

Total labellings, irregularity strengths and colourings

Let G = (V,E) be a simple graph with vertex set V (G) = {v1,v2,...,vn} and edge set E(G). In this paper, first some sharp upper and lower bounds on the largest and least eigenvalues of graphs are given when vertices are removed. Some conjectures in (M. Aouchiche. Comparaison Automatisee d'Invariants en Theorie des Graphes. Ph.D. Thesis, ´ Polytechnique de Montreal, February 2006.) and (M. Aouchiche, G. Caporossi, and P. Hansen. Variable neighborhood search for extremal graphs, 20. Automated comparison of graph invariants. MATCH Commun. Math. Comput. Chem., 58:365-384, 2007.) involving the spectral radius, diameter and matching number are also proved. Furthermore, the extremal graph which attains the minimum least eigenvalue among all quasi-tree graphs is characterized.

Let G = (V, E) be a finite and simple graph with vertex set V (G) and edge set E(G), having order p and size q. A graph G is called super edge-magic if there exists a bijection f : V(G) ∪ E(G) → {1, 2, …, p + q} such that f (V(G)) = {1, 2, …, p} and f (u) + f (uv) + f (v) = k, for every edge uv ∈ E(G). The super edge-magic deficiency of a graph G, denoted by μs(G), is either the minimum nonnegative integer n such that G ∪ nK1 is super edge-magic or +∞ if there exists no such n. In this paper, we study the super edge-magic deficiency of forests where its components are subdivided stars or paths.

Fractional matching preclusion numbers of Cartesian product graphs

Let \(G\) be a simple graph without isolated vertices with vertex set \(V(G)\) and edge set \(E(G)\) and let \(k\) be a positive integer. A function \(f:E(G)\longrightarrow \{-1, 1\}\) is said to be a signed star \(k\)-dominating function on \(G\) if \(\sum_{e\in E(v)}f(e)\ge k\) for every vertex \(v\) of \(G\), where \(E(v)=\{uv\in E(G)\mid u\in N(v)\}\). A set \(\{f_1,f_2,\ldots,f_d\}\) of signed star \(k\)-dominating functions on \(G\) with the property that \(\sum_{i=1}^df_i(e)\le k\) for each \(e\in E(G)\), is called a signed star \((k,k)\)-dominating family (of functions) on \(G\). The maximum number of functions in a signed star \((k,k)\)-dominating family on \(G\) is the signed star \((k,k)\)-domatic number of \(G\), denoted by \(d^{(k,k)}_{SS}(G)\). In this paper we study properties of the signed star \((k,k)\)-domatic number \(d_{SS}^{(k,k)}(G)\). In particular, we present bounds on \(d_{SS}^{(k,k)}(G)\), and we determine the signed \((k,k)\)-domatic number of some regular graphs. Some of our results extend these given by Atapour, Sheikholeslami, Ghameslou and Volkmann [Signed star domatic number of a graph, Discrete Appl. Math. 158 (2010), 213-218] for the signed star domatic number.