If there is a function f : E(G) → {1, 2, 3, . . . , K} suchthat for any two adjacent vertices u and v of the samedegree, the sets S(u) and S(v) are equal, where S(u) ={f(uv), uv ∈ E(G)}, then a graph G = (V, E) is said tohave an adjacent vertex reducible edge coloring. The term“chromatic number of adjacent vertex reducible edge coloring” refers to the highest value of K. The purpose of this study is to ascertain the chromatic number for adjacent vertex reducible edge coloring of specific graphs, including mirror graphs, alternate triangular cycles, prism wheel graphs, double wheel graphs, antiprism graphs, crown (or sun) graphs, degree splitting graphs, and H-graph.

Let be a graph derived from a simple graph G by adding a self-loop to each vertex in a subset . In this paper, we define the atom bond connectivity index of the graph as and the atom bond connectivity energy of as . We obtained upper bounds for the ABC spectral radius of the graph as well as bounds for and in terms of m, n, and . Additionally, we computed the ABC energy for complete graph, cocktail party graph and crown graph with self-loops. We also derived the characteristic polynomial of double star graph with self-loops. Furthermore, we explored the correlation between and various physico-chemical properties, such as boiling point (BP) and molar refraction (MR). Furthermore, we established correlations between and specific indices, specifically the Sombor index of a graph , first Zagreb index of a graph , and Randic index of a graph .

Center function identifies vertices on a graph which minimizes the maximum distance to a set of input vertices. Axiomatic approach attempts to uniquely distinguish location functions such as center, median and mean functions using a set of axioms. In this paper, an attempt is made to find an axiomatic characterization of center function on crown graphs using both universal and nonuniversal axioms.

Hajos Fuzzy graph is a new fuzzy graph obtained by applying a binary operation, named Hajos construction, on two fuzzy graphs. The Hajos construction on two (fuzzy) graphs produces many different (fuzzy) graphs depending on the choice of vertices and edges. The cardinality of Hajos (fuzzy) graphs is the total number of Hajos (fuzzy) graphs from two given Hajos (fuzzy) graphs. Here, the cardinality of Hajós fuzzy graphs are determined for any two fuzzy graphs based on the permutations and combinations method.is By this concept, the cardinality of Hajós (fuzzy) graphs is derived for the combinations of the two (fuzzy) graphs, such as the fan graph, lollipop graph, friendship graph, tadpole graph and crown graphs.

The domination integrity DI(G) of a simple connected graph G is defined as DI(G) = min{| X | +m(G−X) : X is a dominating set } where m(G−X) is the order of a maximum connected component of G − X. It is a measure of vulnerability of a graph. This work is aimed to discuss domination integrity of middle graph of crown graph, middle graph of tadpole graph, middle graph of friendship graph, christmas Star Graph and kusudama flower graph.

Objectives: Laceability partition dimension of a connected graph is the minimum number of partitions of vertex set such that the subgraph induced by each partition is laceable in the case of a bipartite graph and random Hamiltonian laceable in case of non-bipartite graph . Methods: Mathematical Induction method and tracing Hamiltonian path. Findings: This study presents the laceability partition dimension (lpd) of some special graphs namely the Crown graph, Windmill graph, Dutch windmill graph, Cocktail party graph, shadow graph, and image graph. Novelty: This study discusses the laceability partition dimension of the graphs through which we discover a Hamiltonian path within a smaller structure whenever Hamiltonian laceability doesn't exist in large networks which has significance in communication networking that strives for optimal message routing efficiency. Keywords: Hamiltonian path, Bipartite graph, Laceability, Perfect matching, Shadow distance graph

A topological representation of a molecule is called molecular graph. A molecular graph is a collection of points representing the atoms in the molecule and set of lines represent the covalent bonds. Topological indices gather data from the graph of molecule and help to foresee properties of the concealing molecule. All the degree based topological indices have been defined through classical degree concept. In this paper, we define a novel degree concept for a vertex of a simple connected graph: Extended Reverse R degree and also, we define Extended Reverse R indices of a simple connected graph by using the Extended Reverse R degree concept. We compute the Extended Reverse R indices using the above contemporary degree concept for well-known simple connected graphs such as complete bipartite graph, Wheel graph, Generalized Peterson graph, Crown graph, Double star graph, and Windmill graph.

In graph theory, the notion of graph coloring plays an important role and has several applications in the fields of science and engineering. Since the concept of map coloring was first proposed, many researchers have invented a wide range of graph coloring techniques, among which are vertex coloring, edge coloring, total coloring, perfect coloring, list coloring, acyclic coloring, strong coloring, radio coloring, and rank coloring, these are some of the important graph coloring methods that color the graph's vertices, edges, and regions with certain conditions. One of the coloring method is Incident Vertex PI coloring. This is a function of coloring from the set of pairs of incident vertices of every edge of a graph to the power set of colors. This method ensures that all vertices are properly colored, with an additional condition that ordered pair vertices for all edges of graph receive distinct colors. Many types of graphs are defined in the graph theory. In this paper, we have discussed the Incident Vertex PI Coloring numbers for the class of graph families, Fan graph, Book graph, Gear graph, Windmill graph, Dutch Windmill graph and Crown graph.

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.

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.

If the vertices of a graph Γ(α, β), with α = |V(Γ)| and β = |E(Γ)|, can be labelled with unequal integers within the set [0, 2β – 1], so that the edges labels generated by the sum of the labels of the end vertices modulo 2β are distinct odd numbers from the set [1, 2β – 1]. Then, we regard the graph Γ(V, E) to be a semi-odd harmonic graph. An odd harmonious graph satisfies the additional condition that modular asthmatic is not done. A strong od harmonious graph is defined as an odd harmonious graph whose vertices can be labelled with unequal integers from the set [0, β]. In this study, we introduce semi-odd harmonious labeling, which is an extension of odd harmonious labeling. We establish the existence of semi-odd harmonious labeling for the super subdivision of the following graphs: Y–tree, the graph Tg, g–2, the graph Pg  K1, as well as the twig and crown graph. We demonstrate odd harmonious labeling for the super subdivision of the subsequent graphs: cycle, dragon graph Pm(Cg), wheel graph, friendship graph Frg, dumbbell graph Dbm,g, fan graph and the helm graph. The presence of strong odd harmonious labeling for the super subdivision is shown in the final section.

. 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.

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.

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.

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 in E, f*(uv)= ii) |ef*(0)-ef*(1)|<=1where 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 viz., (i) paths (ii) cycles (iii) comb graphs (iv) ladder graphs (v) twig graphs (vi) helm graphs (vii) wheel graphs (viii) crown graphs (ix) star 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.

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.

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 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.

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.