Cartesian Sum Research Articles

The Clustering Coefficient for Graph Products

The clustering coefficient of a vertex v, of degree at least 2, in a graph Γ is obtained using the formula C(v)=2t(v)deg(v)(deg(v)−1), where t(v) denotes the number of triangles of the graph containing v as a vertex, and the clustering coefficient of Γ is defined as the average of the clustering coefficient of all vertices of Γ, that is, C(Γ)=1|V|∑v∈VC(v), where V is the vertex set of the graph. In this paper, we give explicit expressions for the clustering coefficient of corona and lexicographic products, as well as for the Cartesian sum; such expressions are given in terms of the order and size of factors, and the degree and number of triangles of vertices in each factor.

Optimally selecting the top k values from X + Y with layer-ordered heaps.

Selection and sorting the Cartesian sum, X + Y, are classic and important problems. Here, a new algorithm is presented, which generates the top k values of the form . The algorithm relies on layer-ordered heaps, partial orderings of exponentially sized layers. The algorithm relies only on median-of-medians and is simple to implement. Furthermore, it uses data structures contiguous in memory, cache efficient, and fast in practice. The presented algorithm is demonstrated to be theoretically optimal.

Topological Indices and f-Polynomials on Some Graph Products

We obtain inequalities involving many topological indices in classical graph products by using the f-polynomial. In particular, we work with lexicographic product, Cartesian sum and Cartesian product, and with first Zagreb, forgotten, inverse degree and sum lordeg indices.

Harmonic Index and Harmonic Polynomial on Graph Operations

Some years ago, the harmonic polynomial was introduced to study the harmonic topological index. Here, using this polynomial, we obtain several properties of the harmonic index of many classical symmetric operations of graphs: Cartesian product, corona product, join, Cartesian sum and lexicographic product. Some upper and lower bounds for the harmonic indices of these operations of graphs, in terms of related indices, are derived from known bounds on the integral of a product on nonnegative convex functions. Besides, we provide an algorithm that computes the harmonic polynomial with complexity O ( n 2 ) .

Gromov Hyperbolicity in the Cartesian Sum of Graphs

In this paper, we characterize the hyperbolic product graphs for the Cartesian sum $$G_1\oplus G_2$$ : $$G_1\oplus G_2$$ is always hyperbolic, unless either $$G_1$$ or $$G_2$$ is the trivial graph (the graph with a single vertex); if $$G_1$$ or $$G_2$$ is the trivial graph, then $$G_1\oplus G_2$$ is hyperbolic if and only if $$G_2$$ or $$G_1$$ is hyperbolic, respectively. Besides, we characterize the Cartesian sums with hyperbolicity constant $$\delta (G_1\oplus G_2) = t$$ for every value of t. Furthermore, we obtain the sharp inequalities $$1\le \delta (G_1\oplus G_2)\le 3/2$$ for every non-trivial graphs $$G_1,G_2$$ . In addition, we obtain simple formulas for the hyperbolicity constant of the Cartesian sum of many graphs. Finally, we prove the inequalities $$3/2\le \delta (\overline{G_1\oplus G_2})\le 2$$ for the complement graph of $$G_1\oplus G_2$$ for every $$G_1,G_2$$ with $$\min \{{{\mathrm{diam}}}V(G_1), {{\mathrm{diam}}}V(G_2)\}\ge 3$$ .

On the Strong Metric Dimension of Cartesian Sum Graphs

A vertex $w$ of a connected graph $G$ 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 tight bounds or closed formulae for the strong metric dimension of the Cartesian sum of graphs in terms of the strong metric dimension, clique number or twins-free clique number of its factor graphs.

Coloring the Cartesian sum of graphs

For graphs G and H , let G ⊕ H denote their Cartesian sum. We investigate the chromatic number and the circular chromatic number for G ⊕ H . It has been proved that for any graphs G and H , χ ( G ⊕ H ) ≤ max { ⌈ χ c ( G ) χ ( H ) ⌉ , ⌈ χ ( G ) χ c ( H ) ⌉ } . It has been conjectured that for any graphs G and H , χ c ( G ⊕ H ) ≤ max { χ ( H ) χ c ( G ) , χ ( G ) χ c ( H ) } . We confirm this conjecture for graphs G and H with special values of χ c ( G ) and χ c ( H ) . These results improve previously known bounds on the corresponding coloring parameters for the Cartesian sum of graphs.

The [formula omitted]-labeling on Cartesian sum of graphs

An L ( 2 , 1 ) -labeling of a graph G is a function f from the vertex set V ( G ) to the set of all nonnegative integers such that | f ( x ) − f ( y ) | ≥ 2 if d ( x , y ) = 1 and | f ( x ) − f ( y ) | ≥ 1 if d ( x , y ) = 2 , where d ( x , y ) denotes the distance between x and y in G . The L ( 2 , 1 ) -labeling number λ ( G ) of G is the smallest number k such that G has an L ( 2 , 1 ) -labeling with max { f ( v ) : v ∈ V ( G ) } = k . Griggs and Yeh conjecture that λ ( G ) ≤ Δ 2 for any simple graph with maximum degree Δ ≥ 2 . This paper considers the graph formed by the Cartesian sum of two graphs. As corollaries, the new graph satisfies the above conjecture (with minor exceptions).

Coloring the Cartesian Sum of Graphs

Abstract For graphs G and H, let G ⊕ H denote their Cartesian sum. We prove that for any graphs G and H, χ ( G ⊕ H ) ⩽ max { ⌈ χ c ( G ) χ ( H ) ⌉ , ⌈ χ ( G ) χ c ( H ) ⌉ } . Moreover the bound is sharp. We conjecture that for any graphs G and H, χ c ( G ⊕ H ) ⩽ max { χ ( H ) χ c ( G ) , χ ( G ) χ c ( H ) } . The conjectured bound would be sharp if it is true. We confirm this conjecture for graphs G and H with special values of χ c ( G ) and χ c ( H ) . These results improve previously known bounds on the chromatic number and the circular chromatic number for the Cartesian sum of graphs.

Space-time trade-offs for some ranking and searching queries

We study space/time tradeoffs for querying some combinatorial structures. In the first, given an arrangement of n lines in general position in the plane, a query for a real number t asks about Rank(t), the number of vertices of the arrangement with x-coordinates ⩽t. We show that for K= O(n/ logn) , after a preprocessing step that uses space S= O(n 2/(K logK)) the query can be answered in time O(n logK) . The second query involves the Cartesian sum of vectors a=(a 1,…,a n) and b=(b 1,…,b n) . For a given real t, it asks about Rank(t), the number of sums a i+b j which are ⩽t. We show that for some positive constant c and K⩽c( logn)/( log logn) , after a preprocessing step that uses space S= O(n 2/K 2) , the query may be answered in time O((n/K) logK) . Both results fit neatly between two obvious extremes.

A Selection Algorithm for X + Y on Mesh

This paper presents a selection algorithm for finding the k-th largest element in the cartesian sum of two sets X and Y, each of size n, on a mesh connected model. The algorithm has a time complexity of [Formula: see text], using P processors, where P < n2. The algorithm is adaptive but [Formula: see text] away from cost optimality.

Generating Bézier points for curves and surfaces from boundary information

This paper presents efficient methods for directly generating Bézier points of curves and surfaces explicitly from the given compatible arbitrary order boundary information of Hermite curves, Coons-Hermite Cartesian sum patches and Coons-Boolean sum patches. The explicit expressions for the generalized Hermite functions are also developed. Furthermore, a method for determining the twist control points and higher level sets of interior control points from their boundary and lower level sets of control points by using the Coons-Boolean sum schema presented. Many interesting and useful examples are also given in this paper.

On the chromatic number of the lexicographic product and the Cartesian sum of graphs

Let G[ H] be the lexicographic product of graphs G and H and let G ⊕ H be their Cartesian sum. It is proved that if G is a nonbipartite graph, then for any graph H, χ( G( H])⩾2χ( H)+⌜ χ( H)/ k ⌝, where 2 k+1 is the length of a shortest odd cycle of G. Chromatic numbers of the Cartesian sum of graphs are also considered. It is shown in particular that for χ-critical and not complete graphs G and H, χ( G ⊕ H)⩽ χ( G)χ( H) − 1. These bounds are used to calculate chromatic numbers of the Cartesian sum of two odd cycles. Finally, a connection of some colorings with hypergraphs is given.

L-infinity interdistance selection by parametric search

The selection problem asks for the kth largest or smallest element in a set S. In general, selection takes linear time, but if the set is constrained so that some relations between elements are known, sublinear-time selection is sometimes possible. We present an algorithm that selects the kth largest or smallest element from a Cartesian sum in O ( n log n) time and extend this result to select L ∞-interdistances in R d in O( d n log d−1 n) time. The algorithm is based on a general technique o f Megiddo and improved by Cole, and it is the first algorithm for a multidimensional interdistance selection problem with an O( n log O(1) n ) running time.

Selection in X + Y and matrices with sorted rows and columns

Let A be an n×n matrix of reals with sorted rows and columns and k an integer, 1 ⩽ k ⩽ n 2. We present an O(n) time algorithm for selecting the k th smallest element of A. If X and Y are sorted n-vectors of reals, then Cartesian sum X + Y is such a matrix as A. One application of selection in X + Y can be found in statistics. The algorithm presented here is based on a new divide-and-conquer technique, which can be applied to similar order related problems as well. Due to the fact that the algorithm has a relatively small constant time factor, this result is of practical as well as theoretical interest.

Decomposition de K m+ K n en cycles hamiltoniens

Let K m be the complete graph of order m. We prove that the cartesian sum K m + K n can be decomposed into 1 2 ( m+ n−2) hamiltonian cycles if m+ n is even and into 1 2 ( m+ n−3) hamiltonian cycles and a perfect matching if m+ n is odd.

Alcuni risultati sulla teoria degli ipergrafi

We extend the concept ofk-edge-connection from the graph theory (TG) to the hypergraph theory (TH); then, we pass from the definition of Grundy function in TG to its extension in TH; after a suitable definition of kernel in TH, we prove propositions analogous to those linking the two concepts in TG. So, after defining suitable operations of product, cartesian sum, cartesian product of hypergraphs, it is possible to prove several properties due to the Grundy function in TH.

Endomorphism Semigroups of Sums of Rings

Let R= 〈R, +, •〉 be the cartesian sum of the rings Ri, i= 1, 2,…, n denoted by , and recall that R is a ring under the componentwise operations. It is well-known (e.g. [1], p. 212) that the endomorphisms of the group 〈R, + 〉 form a ring HomZR (under function addition and composition) and moreover HomZR is isomorphic to the matrix ring under the usual matrix operations of addition and multiplication.

Chromatic Number of Cartesian Sum of Two Graphs

Chromatic Number of Cartesian Sum of Two Graphs

Chromatic number of Cartesian sum of two graphs

PROOF. If G is a graph, we denote by V(G) the set of vertices of G and by E(G) the set of edges of G. We say that a subset S C V(G) is independent if for any a, b in S, (a, b) ErE(G). If S is a set, we denote by I SI the number of elements in S. Now, let SiC V(Gi) (i= 1, 2) be independent sets such that Sil =3(Gi). If S= {ailDa2: a,ES1 and a2 S2 }, then S is an independent set of G1 E G2. Therefore, 3(GC1 @ G2) > I sI =3(G1)3(G2). Suppose TC V(G1CG2) is an independent set such that I TJ ==13(G1 ED G2). For a G V(Gi), let T(a) ={ b G V(G2) : a0b ET}. Then for each a G V(G1), T(a) is an independent set of G2. Therefore, for each aE V(G1), I T(a) I ?fl(G2). Clearly, I TJ = ZaEV(Gi) I T(a) I. But I T(a) I =0 except for those a in an independent set of G1. Hence j3(GC1 E G2) = 1 TJ <l(GC1)f3(G2). This shows that fl(GC1 C G2) =0(GC1)i(G2). LEMMA 2. K(G1)K(G2) K(CK(G1GC2) .