Complementary Prism Research Articles

Eccentric connectivity index in complementary prisms

The topological index is just one of several very useful tools that graph theory has made available to chemists. Topological indices are invariants of real numbers under graph isomorphisms. Several topological indices have been defined. Some of them are used to model chemical, pharmaceutical and other properties of molecules. The eccentric connectivity index (eci) is also a topological index. The eci of [Formula: see text], denoted by [Formula: see text], is defined as [Formula: see text], where [Formula: see text] represents the degree of a vertex [Formula: see text] and [Formula: see text] is its eccentricity. In this paper, exact formula for the eci of complementary prisms is derived.

Polynomial Representation and Degree Sequence of Graphs Resulting from Some Graph Operations

Let $G = (V(G),E(G))$ be a graph with degree sequence $\langle d_1,d_2, \cdots, d_n \rangle$, where $d_1 \ge d_2 \ge \cdots \ge d_n$. The polynomial representation of $G$ is given by $f_G(x) = \displaystyle \sum_{i=1}^n x^{d_i} = \sum_{k=1}^{\Delta(G)}a_kx^{k}$, where $a_k$ is the number of vertices of $G$ having degree $k$ for each $i = 1,2,\cdots n = \Delta(G)$. In this paper, we give the polynomial representation of the complement and line graph of a graph, the shadow graph, complementary prism, edge corona, strong product, symmetric product, and disjunction of two graphs.

Unique Response Roman Domination Versus 2-Packing Differential in Complementary Prisms

Unique Response Roman Domination Versus 2-Packing Differential in Complementary Prisms

Complexity results on k-independence in some graph products

For a positive integer k, a subset S of vertices of a graph G is k-independent if each vertex in S has at most k − 1 neighbors in S. We consider k-independent sets in two graph products: Cartesian and complementary prism. We show that the problem of determining k-independence remains NP-complete even for Cartesian products and complementary prisms. Furthermore, we present results on k-independence in the Cartesian product of two paths, known as grid graphs.

Convex Roman Dominating Functions on Graphs under some Binary Operations

Let $G$ be a connected graph. A function $f:V(G)\rightarrow \{0,1,2\}$ is a \textit{convex Roman dominating function} (or CvRDF) if every vertex $u$ for which $f(u)=0$ is adjacent to at least one vertex $v$ for which $f(v)=2$ and $V_1 \cup V_2$ is convex. The weight of a convex Roman dominating function $f$, denoted by $\omega_{G}^{CvR}(f)$, is given by $\omega_{G}^{CvR}(f)=\sum_{v \in V(G)}f(v)$. The minimum weight of a CvRDF on $G$, denoted by $\gamma_{CvR}(G)$, is called the \textit{convex Roman domination number} of $G$. In this paper, we specifically study the concept of convex Roman domination in the corona and edge corona of graphs, complementary prism, lexicographicproduct, and Cartesian product of graphs.

Design method for eliminating spectral line tilt in a multiple sub-pupil ultra-spectral imager (MSPUI).

A multiple sub-pupil ultra-spectral imaging system designed with a single spectrometer and detector can simultaneously detect multiple-channel spectra with ultra-high spectral resolution. However, due to using a prism in the system's front end, the nonlinear dispersion introduces spectral line tilt in the imaging spectra. This phenomenon can lead to bias in the final spectral data. To eliminate this issue, we propose a new design by introducing a second prism to correct this spectral tilt in the system. The angle of spectral line tilt generated by the nonlinear dispersion of the first prism is derived. It provides the theoretical basis for characterizing the second complementary prism. Finally, a UV multiple sub-pupil ultra-spectral imaging system is designed. The system employs two pupil separation prisms and one flat panel array to segment the pupil in three channels, each operating within spectral ranges of 180∼210 nm, 275∼305 nm, and 370∼400 nm, respectively. The spectral resolutions in all three channels are better than 0.1 nm. The corrected spectral line tilt is less than 1/3 of a pixel in the two channels with pupil separation prisms. At a Nyquist frequency of 30 lp/mm, the modulation transfer functions of all three channels are greater than 0.7, ensuring imaging quality. The design results indicate that the method proposed in this paper, utilizing complementary prisms, can effectively correct the spectral line tilt caused by the nonlinear dispersion of the pupil separation prisms. This design approach can be a reference for developing multiple sub-pupil ultra-spectral imaging systems.

Transversal total hop domination in graphs

Let [Formula: see text] be a finite simple graph. A set [Formula: see text] is a total hop dominating set of [Formula: see text] if for every [Formula: see text], there exists [Formula: see text] such that [Formula: see text]. Any total hop dominating set of [Formula: see text] of minimum cardinality is a [Formula: see text]-set of [Formula: see text]. A total hop dominating set [Formula: see text] of [Formula: see text] which intersects every [Formula: see text]-set of [Formula: see text] is a transversal total hop dominating set. The minimum cardinality [Formula: see text] of a transversal total hop dominating set in [Formula: see text] is the transversal total hop domination number of [Formula: see text]. In this paper, we initiate the study of transversal total hop domination in graphs. First, we characterize a graph [Formula: see text] of order [Formula: see text] for which [Formula: see text] is [Formula: see text] or [Formula: see text], and also we determine the specific values of [Formula: see text] for some special graphs [Formula: see text]. Next, we solve some realization problems involving [Formula: see text] with other parameters of [Formula: see text]. Finally, we investigate the transversal total hop domination in the complementary prism, corona, and lexicographic product of graphs.

On the monophonic convexity in complementary prisms

A set S of vertices of a graph G is monophonic convex if S contains all the vertices belonging to any induced path connecting two vertices of S. The cardinality of a maximum proper monophonically convex set of G is called the monophonic convexity number of G. The monophonic interval of a set S of vertices of G is the set S together with every vertex belonging to any induced path connecting two vertices of S. The cardinality of a minimum set S⊆V(G) whose monophonic interval is V(G) is called the monophonic interval number of G. The monophonic convex hull of a set S of vertices of G is the smallest monophonically convex set containing S in G. The cardinality of a minimum set S⊆V(G) whose monophonic convex hull is V(G) is called the monophonic hull number of G. The complementary prismGG¯ of G is obtained from the disjoint union of G and its complement G¯ by adding the edges of a perfect matching between them. In this work, we show that the corresponding decision problem to the monophonic convexity number is NP-complete even for complementary prisms. On the other hand, we show that the monophonic interval number and the monophonic hull number can be determined in linear time for the complementary prisms of all graphs.

The core of a vertex transitive complementary prism of a lexicographic product

The complementary prism of a graph Γ is the graph ΓΓ̄, which is formed from the union of Γ and its complement Γ̄ by adding an edge between each pair of identical vertices in Γ and Γ̄. Vertex-transitive self-complementary graphs provide vertex-transitive complementary prisms. It was recently proved by the author that ΓΓ̄ is a core, i.e. all its endomorphisms are automorphisms, whenever Γ is vertex-transitive, self-complementary, and either Γ is a core or its core is a complete graph. In this paper the same conclusion is obtained for some other classes of vertex-transitive self-complementary graphs that can be decomposed as a lexicographic product Γ = Γ1[Γ2]. In the process some new results about the homomorphisms of a lexicographic product are obtained.

Pointwise Clique-Safe Domination in the Complement and Complementary Prism of Special Families of Graphs

Pointwise Clique-Safe Domination in the Complement and Complementary Prism of Special Families of Graphs

Closeness Centrality of Vertices in Graphs Under Some Operations

In this paper, we revisit the concept of (normalized) closeness centrality of a vertex in a graph and investigate it in some graphs under some operations. Specifically, we derive formulas that compute the closeness centrality of vertices in the shadow graph, complementary prism, edge corona, and disjunction of graphs.

The core of a complementary prism

The complementary prism Gamma {bar{Gamma }} is constructed from the disjoint union of a graph Gamma and its complement {bar{Gamma }} if an edge is added between each pair of identical vertices in Gamma and {bar{Gamma }}. It generalizes the Petersen graph, which is obtained if Gamma is the pentagon. The core of the complementary prism is investigated for arbitrary simple graph Gamma . In particular, it is shown that if Gamma is strongly regular and self-complementary, then Gamma {bar{Gamma }} is a core, i.e. all its endomorphisms are automorphisms.

The core of a vertex-transitive complementary prism

The complementary prism ΓΓ̄ is obtained from the union of a graph Γ and its complement Γ̄ where each pair of identical vertices in Γ and Γ̄ is joined by an edge. It generalizes the Petersen graph, which is the complementary prism of the pentagon. The core of a vertex-transitive complementary prism is studied. In particular, it is shown that a vertex-transitive complementary prism ΓΓ̄ is a core, i.e. all its endomorphisms are automorphisms, whenever Γ is a core or its core is a complete graph.

On Double Roman Dominating Functions in Graphs

Let G be a connected graph. A function f : V (G) → {0, 1, 2, 3} is a double Roman dominating function of G if for each v ∈ V (G) with f(v) = 0, v has two adjacent vertices u and w for which f(u) = f(w) = 2 or v has an adjacent vertex u for which f(u) = 3, and for each v ∈ V (G) with f(v) = 1, v is adjacent to a vertex u for which either f(u) = 2 or f(u) = 3. The minimum weight ωG(f) = P v∈V (G) f(v) of a double Roman dominating function f of G is the double Roman domination number of G. In this paper, we continue the study of double Roman domination introduced and studied by R.A. Beeler et al. in [2]. First, we characterize some double Roman domination numbers with small values in terms of the domination numbers and 2-domination numbers. Then we determine the double Roman domination numbers of the join, corona, complementary prism and lexicographic product of graphs.

Geometric and topological properties of the complementary prism networks

The complementary prism of , denoted by , is the graph obtained from the disjoint union of and by adding edges between the corresponding vertices of and . In this paper, we study the hyperbolicity constant of . In particular, we obtain upper and lower bounds for the hyperbolicity constant, and we compute its precise value for many graphs. Moreover, we obtain bounds and closed formulas for the general topological indices and (where denotes the degree of the vertex is a symmetric function with real values, and is a function with real values), and the generalized Wiener index , of complementary prisms networks. Finally, we performed a numerical study of the generalized Wiener index on random graphs.

Study on (r + 1)-role assignments of complementary prisms, with r ≥ 3

Study on (r + 1)-role assignments of complementary prisms, with r ≥ 3

Complexity results on open-independent, open-locating-dominating sets in complementary prism graphs

For a finite, simple, undirected graph G = ( V ( G ) , E ( G ) ) , an open-dominating set S ⊆ V ( G ) is such that every vertex in G has at least one neighbor in S . An open-independent, open-locating-dominating set S ⊆ V ( G ) ( O L D o i n d -set for short) is such that 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 such set is known to be NP -complete. The complementary prism of G is the graph G G ¯ , formed from the disjoint union of G and its complement G ¯ by adding the edges of a perfect matching between the corresponding vertices of G and G ¯ . Various properties, upper bounds and logarithmic lower bounds on the sizes of minimal O L D o i n d -sets in complementary prisms are presented. We show that deciding, for a given graph G , whether or not G G ¯ has an O L D o i n d -set is an NP -complete problem. On the other hand, if the girth of G is at least four, we show that the O L D o i n d -set of G G ¯ , if it exists, can be found in polynomial time.

A note on the convexity number of the complementary prisms of trees

A set of vertices S of a graph G is a (geodesically) convex set, if S contains all the vertices belonging to any shortest path connecting two vertices of S . The cardinality of a maximum proper convex set of G is called the convexity number, con ( G ) , of G . The complementary prism G G ¯ of G is obtained from the disjoint union of G and its complement G ¯ by adding the edges of a perfect matching between them. In this work, we examine the convex sets of the complementary prism of a tree and derive formulas for the convexity numbers of the complementary prisms of all trees.

A polynomial time algorithm for geodetic hull number for complementary prisms

Let G be a finite, simple, and undirected graph and let S ⊆ V (G). In the geodetic convexity, S is convex if all vertices belonging to any shortest path between two vertices of S lie in S. The convex hull H(S) of S is the smallest convex set containing S. The hull number h(G) is the minimum cardinality of a set S ⊆ V (G) such that H(S) = V (G). The complementary prism GG̅ of a graph G arises from the disjoint union of the graph G and G̅ by adding the edges of a perfect matching between the corresponding vertices of G and G̅. Previous works have determined h(GG̅) when both G and G̅ are connected and partially when G is disconnected. In this paper, we characterize convex sets in GG̅ and we present equalities and tight lower and upper bounds for h(GG̅). This fills a gap in the literature and allows us to show that h(GG̅) can be determined in polynomial time, for any graph G.

K-Independence on complementary prism graphs

k-Independence on complementary prism graphs