Finite Simple Graph Research Articles

Rough sets on graphs: New dependency and accuracy measures

Attribute dependency and approximation accuracy measures are two fundamental notions in rough set and database theory. In this paper, we provide combinatorial closed formulas and asymptotic estimates for the attribute dependency and the approximation accuracy of some basic graph families. Actually, a finite simple undirected graph can be seen as a particular case of information table. This perspective can be assumed in a not unique way: in fact, in terms of information table, we can investigate a connected graph by means of its adjacency matrix or through its minimum vertex distance matrix. Having assumed both these two perspectives, we study the behavior of the attribute dependency and the approximation accuracy for some basic graph families and, above all, we provide an asymptotic estimate for two global averages obtained by means of the previous quantities.

Reflexive polytopes arising from edge polytopes

It is known that every lattice polytope is unimodularly equivalent to a face of some reflexive polytope. A stronger question is to ask whether every (0,1)-polytope is unimodularly equivalent to a facet of some reflexive polytope. A large family of (0,1)-polytopes are the edge polytopes of finite simple graphs. In the present paper, it is shown that, by giving a new class of reflexive polytopes, each edge polytope is unimodularly equivalent to a facet of some reflexive polytope. Furthermore, we extend the characterization of normal edge polytopes to a characterization of normality for these new reflexive polytopes.

Critical Paired Dominating Sets and Irreducible Decompositions of Powers of Edge Ideals

Let G be a finite simple graph. A set S of vertices is a critical paired dominating set of G, if every vertex is adjacent to a vertex in S and the removal of any vertex does not change the matching number of G. In this paper, we give a characterization of graphs G which has a critical paired dominating set in terms of the irreducible decomposition of powers of the edge ideal associated to G.

On the structure of $S_2$-ifications of complete local rings

Motivated by work of Hochster and Huneke, we investigate several constructions related to the $S_2$-ification $T$ of a complete equidimensional local ring $R$: the canonical module, the top local cohomology module, topological spaces of the form $Spec (R)-V(J)$, and the (finite simple) graph $\Gamma _R$ with vertex set $Min (R)$ defined by Hochster and Huneke. We generalize one of their results by showing, e.g., that the number of connected components of $\Gamma _R$ is equal to the maximum number of connected components of $Spec (R)-V(J)$ for all $J$ of height $2$. We further investigate this graph by exhibiting a technique for showing that certain graphs $G$ can be realized in the form $\Gamma _R$.

Embeddability of right-angled Artin groups on complements of trees

For a finite simplicial graph [Formula: see text], let [Formula: see text] denote the right-angled Artin group on [Formula: see text]. Recently, Kim and Koberda introduced the extension graph [Formula: see text] for [Formula: see text], and established the Extension Graph Theorem: for finite simplicial graphs [Formula: see text] and [Formula: see text], if [Formula: see text] embeds into [Formula: see text] as an induced subgraph then [Formula: see text] embeds into [Formula: see text]. In this paper, we show that the converse of this theorem does not hold for the case [Formula: see text] is the complement of a tree and for the case [Formula: see text] is the complement of a path graph.

On Betti numbers of edge ideals of crown graphs

An n-crown $$\mathcal {C}_{n,n}$$ on 2n vertices is a graph obtained from complete bipartite graph $${K}_{n,n}$$ by removing edges of a perfect matching. Given a finite simple graph G, one can associate a simplicial complex $$\Delta (G)$$ . In this paper, we use combinatorial data from the associated simplicial complex $$\Delta (\mathcal {C}_{n,n})$$ of the crown graph $$\mathcal {C}_{n,n}$$ and give a formula to find Betti numbers of the form $$\beta _{i,i+1}$$ of edge ideals of $$\mathcal {C}_{n,n}$$ . We also present a formula to find a particular Betti of the edge ideal of a crown graph. We explicitly compute the projective dimension of the edge ideals of crown graphs using domination parameters of the graphs.

On edge-graceful labeling and deficiency for regular graphs

An edge-graceful labeling of a finite simple graph with p vertices and q edges is a bijection from the set of edges to the set of integers {1,2,…,q} such that the vertex sums are pairwise distinct modulo p, where the vertex sum at a vertex is the sum of labels of all edges incident to such vertex. A graph is called edge-graceful if it admits an edge-graceful labeling. In 2005 Hefetz (2005) proved that a regular graph of even degree is edge-graceful if it contains a 2-factor consisting of mCn, where m,n are odd. In this article, we show that a regular graph of odd degree is edge-graceful if it contains either of two particular 3-factors, namely, a claw factor and a quasi-prism factor. We also introduce a new notion called edge-graceful deficiency, which is a parameter to measure how close a graph is away from being an edge-graceful graph. In particular the edge-graceful deficiency of a regular graph of even degree containing a Hamiltonian cycle is completely determined.

Problems on matchings and independent sets of a graph

Let G be a finite simple graph. For X⊂V(G), the difference of X, d(X)≔|X|−|N(X)| where N(X) is the neighborhood of X and max{d(X):X⊂V(G)} is called the critical difference of G. X is called a critical set if d(X) equals the critical difference and ker(G) is the intersection of all critical sets. diadem(G) is the union of all critical independent sets. An independent set S is an inclusion minimal set withd(S)>0 if no proper subset of S has positive difference.A graph G is called a König–Egerváry graph if the sum of its independence number α(G) and matching number μ(G) equals |V(G)|.In this paper, we prove a conjecture which states that for any graph the number of inclusion minimal independent set S with d(S)>0 is at least the critical difference of the graph.We also give a new short proof of the inequality |ker(G)|+|diadem(G)|≤2α(G).A characterization of unicyclic non-König–Egerváry graphs is also presented and a conjecture which states that for such a graph G, the critical difference equals α(G)−μ(G), is proved.We also make an observation about ker(G) using Edmonds–Gallai Structure Theorem as a concluding remark.

On the complexity of graph coloring with additional local conditions

Let G=(V,E) be a finite simple graph. Recall that a proper coloring of G is a mapping φ:V→{1,…,k} such that every color class induces an independent set. Such a φ is called a semi-matching coloring if the union of any two consecutive color classes induces a matching. We show that the semi-matching coloring problem is NP-complete for any fixed k⩾3, and we get the same result for another version of this problem in which any triangle of G is required to have vertices whose colors differ at least by three.

A note on [formula omitted]-near-factor-critical graphs

A factor (near-factor) of a finite simple graph G is a matching that saturates all vertices (except one). For m⩾0, a graph G is said to be m-critical (m-near-critical) if the deletion of any m vertices from G produces a subgraph that has a factor (near-factor). An m-critical graph is (m+1)-near-critical. The following results are established. (i) Within the class of (m+1)-near-critical graphs, a characterization is given for those that are not m-critical. (ii) For an (m+1)-connected graph, it is (m+1)-near-critical if and only if it is m-critical. (iii) An (m+2)-near-critical graph is m-near-critical if its order is at least m+5.

On super [formula omitted]-[formula omitted]-antimagic total labeling of disjoint union of cycles

Let H and G be finite simple graphs where every edge of G belongs to at least one subgraph that is isomorphic to H. An (a,d)-H-antimagic total labeling of a graph G is a bijection f:V(G)∪E(G)→{1,2,…,|V(G)|+|E(G)|} such that for all subgraphs H′ isomorphic to H, the H-weights, w(H′)=∑v∈V(H′)f(v)+∑uv∈E(H′)f(uv) form an arithmetic progression {a,a+d,…,a+(k−1)d} where a>0,d≥0 are two fixed integers and k is the number of subgraphs of G isomorphic to H. Moreover, if the vertex set V(G) receives the minimum possible labels {1,2,…,|V(G)|}, then f is called a super(a,d)-H-antimagic total labeling. In this paper we study super (a,d)-Cn-antimagic total labeling of a disconnected graph, namely mCn.

A characterisation on arc-transitive graphs of prime valency

Let Γ be a finite simple undirected graph and G ≤ Aut(Γ). If G is transitive on the set of s-arcs but not on the set of (s+1)-arcs of Γ, then Γ is called (G, s)-transitive. For a connected (G, s)-transitive graph Γ of prime valency, the vertex-stabilizer Gα with α ∈ V(Γ) has been determined when Gα is solvable. In this paper, we give a characterization of the vertex-stabilizers of (G, s)-transitive graphs of prime valency when Gα is unsolvable.

On normal graph of a finite group

Suppose G is a finite group and C(G) denotes the set of all conjugacy classes of G. The normal graph of G, N(G), is a finite simple graph such that V(N(G)) = C(G). Two conjugacy classes A and B in C(G) are adjacent if and only if there is a proper normal subgroup N such that A U B ? N. The aim of this paper is to study the normal graph of a finite group G. It is proved, among other things, that the groups with identical character table have isomorphic normal graphs and so this new graph associated to a group has good relationship by its group structure. The normal graphs of some classes of finite groups are also obtained and some open questions are posed.

Global Synchronization of Pulse-Coupled Oscillators on Trees

Consider a distributed network on a finite simple graph $G=(V,E)$ with diameter $d$ and maximum degree $\Delta$, where each node has a phase oscillator revolving on $S^1=\mathbb{R}/\mathbb{Z}$ with unit speed. Pulse-coupling is a class of distributed time evolution rule for such networked phase oscillators inspired by biological oscillators, which depends only upon event-triggered local pulse communications. In this paper, we propose a novel inhibitory pulse-coupling and prove that arbitrary phase configuration on $G$ synchronizes by time $51d$ if $G$ is a tree and $\Delta\leq3$. We extend this pulse-coupling by letting each oscillator throttle the input according to an auxiliary state variable. We show that the resulting adaptive pulse-coupling synchronizes arbitrary initial configuration on $G$ by time $83d$ if $G$ is a tree. As an application, we obtain a universal randomized distributed clock synchronization algorithm, which uses $O(\log\Delta)$ memory per node and converges on any $G$ with an expected worst case running time of $O(|V|+(d^5+\Delta^2)\log|V|)$.

Edgewise strongly shellable clutters

When [Formula: see text] is a chordal clutter in the sense of Woodroofe or Emtander, we show that the complement clutter is edgewise strongly shellable. When [Formula: see text] is indeed a finite simple graph, we provide additional characterization of chordal graphs from the point of view of strong shellability. In particular, the generic graph [Formula: see text] of a tree is shown to be bi-strongly shellable.

Finite Simple Graphs and Their Associated Graph Lattices

Finite Simple Graphs and Their Associated Graph Lattices

K-domination number of products of two directed cycles and two directed paths

Let D be a finite simple directed graph with vertex set V(D) and arc set A(D). A subset S of the vertex set V(D) is a k-dominating set (k ≥ 1) of D if for each vertex v not in S there exists k vertices {ui, ..., uk} ⊆ S such that (ui, v) is an arc of D for i = 1, ..., k. The k-domination number of D, k(D), is the cardinality of the smallest k-dominating set of D. The k-domination number (k ≥ 2) of the Cartesian products of two directed cycles, two directed paths and Cartesian products of a directed path and a cycle are determined. Also, we give k-domination number (k ≥ 2) of the direct product of two directed cycles and two directed paths.

On Edge-Graceful Regular Graphs with Particular 3-Factors

An edge-graceful labeling of a finite simple graph with p vertices and q edges is a bijection from the set of edges to the set of integers {1,2,⋯,q} such that the vertex sums are pairwise distinct modulo p, where the vertex sum at a vertex is the sum of labels of all edges incident to such vertex. A graph is called edge-graceful if it admits an edge-graceful labeling. In this article, we verify that an regular graph of odd degree is edge-graceful if it contains either of two particular 3-regular spanning subgraphs, namely, a quasi-prism factor and a claw factor.

Lower bounds for the energy of graphs

Let G be a finite simple undirected graph with n vertices and m edges. The energy of a graph G, denoted by E(G), is defined as the sum of the absolute values of the eigenvalues of G. In this paper we present lower bounds for E(G) in terms of number of vertices, edges, Randić index, minimum degree, diameter, walk and determinant of the adjacency matrix. Also we show our lower bound in (11) under certain conditions is better than the classical bounds given in Caporossi et al. (1999), Das (2013) and McClelland (1971).

An introduction to the twin signed total k-domination numbers in directed graphs

Let D = (V,A ) be a finite simple directed graph (shortly digraph), N − (v ) and N + (v ) denote the set of in-neighbors and out-neighbors of a vertex v ∈ V , respectively. A function f :V − → { − 1,1 } is called a twin signed total k -dominating function (TSTk DF) if ∑ u ∈ (N − (v ))f (u ) ≥ k and ∑ u ∈ (N + (v ))f (u ) ≥ k for each vertex v ∈ V . The twin signed total k -domination number of D is γ ∗ stk (D ) = min{ω (f ) | f is a TSTk DF of D }, where ω (f ) = ∑ v ∈ V f (v ) is the weight of f . In this paper, we initiate the study of twin signed total k -domination in digraphs and present different bounds on γ ∗ stk (D ). In addition, we determine the twin signed total k -domination number of some classes of digraphs. Our results are mostly extensions of well-known bounds of the twin signed total domination numbers of directed graphs.