Crown Graph Research Articles

Incident Vertex Pi Coloring of Graph Families: Fan, Book, Gear, Windmill, Dutch Windmill and Crown Graph

Incident Vertex Pi Coloring of Graph Families: Fan, Book, Gear, Windmill, Dutch Windmill and Crown Graph

Thinness and its variations on some graph families and coloring graphs of bounded thinness

Interval graphs and proper interval graphs are well known graph classes, for which several generalizations have been proposed in the literature. In this work, we study the (proper) thinness, and several variations, for the classes of cographs, crowns graphs and grid graphs. We provide the exact values for several variants of thinness (proper, independent, complete, precedence, and combinations of them) for the crown graphs CRn. For cographs, we prove that the precedence thinness can be determined in polynomial time. We also improve known bounds for the thinness of n × n grids GRn and m×n grids GRm,n, proving that n−1/3 ≤ thin(GRn) ≤ n+1/2. Regarding the precedence thinness, we prove that prec-thin(GRn,2) = n+1/2 and that n− 1 + 3/2 ≤ prec-thin(GRn) ≤ n− 1 2. As applications, we show that the k-coloring problem is NP-complete for precedence 2-thin graphs and for proper 2-thin graphs, when k is part of the input. On the positive side, it is polynomially solvable for precedence proper 2-thin graphs, given the order and partition.

The Minimum Superior Dominating Energy of Graphs

Kathiresan and Marimuthu were the pioneers of superior distance in graphs. The same authors put forth the concept of superior domination in 2008. Superior distance is the shortest walk between any two vertices including their respective neighbours. The minimum superior dominating energy is defined by the sum of the eigenvalues and it is obtained from the minimum superior dominating matrix . The minimum superior dominating energy for star and crown graphs are computed. Properties of eigenvalues of minimum superior dominating matrix for star, cocktail party, complete and crown graphs are discussed. Results related to upper and lower bounds of minimum superior dominating energy for star, cocktail party, complete and crown graphs are stated and proved.

Group Difference Cordial Labeling in Graphs

. Let group .For
 denotes 
 . 
 which is the group of fourth roots of unity, that is cyclic with generators i and -i. We proved that path graph, bistar graph, and Crown graph are We Ladder graph, and star graph is a difference cordial graph.

Bipartite Domination Number of Mycielski Graph of Some Graph Families

For a nontrivial connected graph G, a non-empty set S \(\subseteq\) V (G) is a bipartite dominating set of graph G, if the subgraph G[S] induced by S is bipartite and for every vertex not in S is dominated by any vertex in S. The bipartite domination number denoted by \(\gamma\)bip(G) of graph G is the minimum cardinality of a bipartite dominating set G. In this paper, we determine the exact bipartite domination number of a crown graph and its mycielski graph as well as the bipartite domination number of the mycielski graph of path and cycle graphs.

Optimal Tilings of Bipartite Graphs Using Self-Assembling DNA

Motivated by the recent advancements in nanotechnology and the discovery of new laboratory techniques using the Watson-Crick complementary properties of DNA strands, formal graph theory has recently become useful in the study of self-assembling DNA complexes. Construction methods based on graph theory have resulted in significantly increased efficiency. We present the results of applying graph theoretical and linear algebra techniques for constructing crossed-prism graphs, crown graphs, book graphs, stacked book graphs, and helm graphs, along with kite, cricket, and moth graphs. In particular, we explore various design strategies for these graph families in two sets of laboratory constraints.

A note on m-Zumkeller cordial labeling of graphs

Let G = (V,E) be a graph. An m-Zumkeller cordial labeling of the graph G is defined by an injective function f : V → N such that there exists an induced function f ∗ : E → {0, 1} defined by f ∗ (uv) = f (u) · f (v) that satisfies the following conditions- i. For every uv ∈ E, f ∗ (uv) = {█(1,if f (u) · f (v) is an m-Zumkeller number @ 0,otherwise )┤ ii. |ef ∗ (0) − ef∗ (1)| ≤ 1 where ef ∗ (0) and ef ∗ (1) denote the number of edges of the graph G having label 0 and 1 respectively under f ∗. In this paper we prove that there exist an m-Zumkeller cordial labeling of graphs in paths, cycles, comb graphs, ladder graphs, twig graphs, helm graphs, wheel graphs, crown graphs and star graphs.

Similar d– even vertex odd mean labeling of diverse graphs

An injective function F where, F : V(G) → {0,2,4, ..., 2b + 2d - 2} is said to be d-even vertex odd mean labeling (d-EVOML) of the graph G(a, b) when the induced mapping F* : E(G) → {1,3, ..., 2b - 1} given by: F*(lw) = F(l)+F(w)/2, is a bijective function. A d - even vertex odd mean graph is a graph which allows even vertex odd mean labelling. In this study, we identify the lowest value of d for which the graphs: Y-tree, star graph, Px ʘ Kh, crown graph Rh, and rooted product Ph◊C4 have a d- even vertex odd mean labeling. Furthermore, we find the minimum number d for which the graphs: cycle graph Ch when h ≡ 2 mod 4, dragon graph Px(Ch) when h ≡ 2 mod 4, x ≥ 1, prism graph Ph, and Toeplitz graphs Th(1, 3), Th(1, 5) and Th(1, 3, 5) have a similar d- even vertex odd mean labeling. In the end, we establish that no odd cycle Ch is an even vertex odd mean graph for all d.

Rings Domination Number of Some Mycielski Graphs

A set S of a graph G = (V (G);E(G)) is a rings dominating set if S is a dominating set and for every vertex in the complement of S has atleast two adjacent vertices. The caridinality of the minimum rings dominating set is the rings domination number of graph G, denoted by \(\gamma\)ri(G). In this paper we determine the exact rings domination number of the mycielski graphs of path graph, cycle graph, and crown graph including its parameter.

Circulant almost cross intersecting families

Let ℱ and G be two t-uniform families of subsets over [k] = {1, 2, ..., k}, where |ℱ| = |G|, and let C be the adjacency matrix of the bipartite graph whose vertices are the subsets in ℱ and G, where there is an edge between A ∈ ℱ and B ∈ G if and only if A ∩ B ≠ ∅. The pair (ℱ,G) is q-almost cross intersecting if every row and column of C has exactly q zeros. We further restrict our attention to q-almost cross intersecting pairs that have a circulant intersection matrix Cp, q, determined by a column vector with p > 0 ones followed by q > 0 zeros. This family of matrices includes the identity matrix in one extreme, and the adjacency matrix of the bipartite crown graph in the other extreme. We give constructions of pairs (ℱ,G) whose intersection matrix is Cp, q, for a wide range of values of the parameters p and q, and in some cases also prove matching upper bounds. Specifically, we prove results for the following values of the parameters: (1) 1 ≤ p ≤ 2t − 1 and 1 ≤ q ≤ k − 2t + 1. (2) 2t ≤ p ≤ t2 and any q > 0, where k ≥ p + q. (3) p that is exponential in t, for large enough k. Using the first result we show that if k ≥ 4t − 3 then C2t − 1, k − 2t + 1 is a maximal isolation submatrix of size k × k in the 0, 1-matrix Ak, t, whose rows and columns are labeled by all subsets of size t of [k], and there is a one in the entry on row x and column y if and only if subsets x, y intersect.

The line graph of the crown graph is distance integral

The distance eigenvalues of a connected graph G are the eigenvalues of its distance matrix . A graph is called distance integral if all of its distance eigenvalues are integers. Let be an integer. A crown graph is a graph obtained from the complete bipartite graph by removing a perfect matching. Let denote the line graph of the crown graph . In this paper, by using the orbit partition method in algebraic graph theory, we determine the set of all distance eigenvalues of and show that this graph is distance integral.

On regular graphs equienergetic with their complements

We give necessary and sufficient conditions on the parameters of a regular graph Γ (with or without loops) such that . We study complementary equienergetic cubic graphs obtaining classifications up to isomorphisms for connected cubic graphs with single loops (5 non-isospectral pairs) and connected integral cubic graphs without loops ( or ). Then we show that, up to complements, the only bipartite regular graphs equienergetic and non-isospectral with their complements are the crown graphs or . Next, for the family of strongly regular graphs Γ we characterize all possible parameters such that . Furthermore, using this, we prove that a strongly regular graph is equienergetic to its complement if and only if it is either a conference graph or else it is a pseudo Latin square graph (i.e. has OA parameters). We also characterize all complementary equienergetic pairs of graphs of type , and in Cameron's hierarchy (the cases in the non-bipartite case and are still open). Finally, we consider unitary Cayley graphs over rings . We show that if R is a finite Artinian ring with an even number of local factors, then is complementary equienergetic if and only if is the product of 2 finite fields.

H- and H2- cordial labeling of graphs in the context of some graph operations on crown graph

H- and H2- cordial labeling of graphs in the context of some graph operations on crown graph

Product cordial graph in the context of some graph operations on crown, helm, and wheel graph

Product cordial graph in the context of some graph operations on crown, helm, and wheel graph

Soft Graphs of Certain Graphs

Let G ∗ = (V, E) be a simple graph and A be any nonempty set of parameters. Let subset R of A×V be an arbitrary relation from A to V. A mapping from A to ᵖ(V) written as F:A → ᵖ(V) can be defined as F(x) = { y ∈ V/xRy} and a mapping from A to ᵖ(E) written as K:A → ᵖ(E) can be defined as K(x) = {uv ∈ E/{u, v} ⊆ F(x)}. The pair (F, A) is a soft set over V and the pair (K, A) is a soft set over E. Obviously (F(a), K(a)) is a subgraph of G ∗ for all a ∈ A. The 4-tuple G = ( G ∗, F, K, A) is called a soft graph of G. In this paper we discuss different soft graphs of graphs such as Complete graph, Star graph, Complete bipartite graph, Crown graph, Comb graph, Friendship graph, Bistar graph and Wheel graph.

Algebraic Properties of Edge Ideals and Cover Ideals of Unbalanced Crown Graphs

In this paper, we introduce the unbalanced crown graphs $${\mathcal {C}}_{m,n}$$ , and compute all graded Betti numbers of edge ideals of these graphs in terms of associated numerical data. As a consequence, we determine the regularity and the projective dimension of edge ideals of these graphs. Also, the Hilbert series of the cover ideals of these graphs is obtained.

Extending of Edge Even Graceful Labeling of Graphs to Strong r -Edge Even Graceful Labeling

Edge even graceful labeling of a graph G with p vertices and q edges is a bijective f from the set of edge E G to the set of positive integers 2,4 , … , 2 q such that all the vertex labels f ∗ V G , given by f ∗ u = ∑ u v ∈ E G f u v mod 2 k , where k = max p , q , are pairwise distinct. There are many graphs that do not have edge even graceful labeling, so in this paper, we have extended the definition of edge even graceful labeling to r -edge even graceful labeling and strong r -edge even graceful labeling. We have obtained the necessary conditions for more path-related graphs and cycle-related graphs to be an r -edge even graceful graph. Furthermore, the minimum number r for which the graphs: tortoise graph, double star graph, ladder and diagonal ladder graphs, helm graph, crown graph, sunflower graph, and sunflower planar graph, have an r -edge even graceful labeling was found. Finally, we proved that the even cycle C 2 n has a strong 2 -edge even graceful labeling when n is even.

On the irregular coloring of bipartite graph and tree graph families

This article discusses irregular coloring. Irregular coloring was first introduced by Mary Radcliffe and Ping Zhang in 2007. The coloring c is called irregular coloring if distinct vertices of G have distinct codes. The color code of a vertex v of G with respect to c is code(v) = (a0, a1, a2,…, a k) = a0a1a2, …ak, where a0 = c(v) and ai, (1 < i < k) is the number of vertices that are adjacent to v and colored i. The minimum k-color used in irregular coloring is called the irregular chromatic number and is denoted by \\ir. Irregular coloring is included in proper coloring, where each vertex that is the neighbors must not be the same color. The graphs used in this article are a family of bipartite graphs and a family of tree graphs, including complete bipartite graphs, crown graphs, star graphs, centipede graphs, and double star graphs.

On the spectral gap and the automorphism group of distance-regular graphs

We prove that a distance-regular graph with a dominant distance is a spectral expander. The key ingredient of the proof is a new inequality on the intersection numbers. We use the spectral gap bound to study the structure of the automorphism group.The minimal degree of a permutation group G is the minimum number of points not fixed by non-identity elements of G. Lower bounds on the minimal degree have strong structural consequences on G. In 2014 Babai proved that the automorphism group of a strongly regular graph with n vertices has minimal degree ≥cn, with known exceptions. Strongly regular graphs correspond to distance-regular graphs of diameter 2. Babai conjectured that Hamming and Johnson graphs are the only primitive distance-regular graphs of diameter d≥3 whose automorphism group has sublinear minimal degree. We confirm this conjecture for non-geometric primitive distance-regular graphs of bounded diameter. We also show if the primitivity assumption is removed, then only one additional family of exceptions arises, the crown graphs. We settle the geometric case in a companion paper.

Symbolic Powers of Certain Cover Ideals of Graphs

In this paper, we compute the regularity and Hilbert series of symbolic powers of the cover ideal of a graph $G$ when $G$ is either a crown graph or a complete multipartite graph. We also compute the multiplicity of symbolic powers of cover ideals in terms of the number of edges.