Odd Number Of Vertices Research Articles

Fuss’ Relation for Bicentric Tridecagon and Bicentric Pentadecagon

This article describes a general analytical derivation of the Fuss’ relation for bicentric polygons with an odd number of vertices. In particular, we derive the Fuss’ relations for the bicentric tridecagon and the bicentric pentadecagon.

On Broadcast Schemes of Knödel Graphs

Knödel graphs on even number of vertices serve as crucial components in constructing various broadcast graphs, particularly those on an odd number of vertices. The efficacy of these constructions heavily relies on the chosen broadcast scheme. Dimensional broadcast schemes are widely employed for broadcasting Knödel graphs. This paper delves into the investigation of standard and reverse dimensional broadcast schemes identifying numerous valid schemes. Specifically, let [Formula: see text] be a Knödel graph of order [Formula: see text] and [Formula: see text]. We explore the impact of [Formula: see text] cyclic shifts on both standard and reverse dimensional broadcast schemes of Knödel graphs, where [Formula: see text], with repetition of the second dimension.

Extremal results on the Mostar index of trees with fixed parameters

For a graph [Formula: see text], the Mostar index of [Formula: see text] is the sum of [Formula: see text] over all edges [Formula: see text] of [Formula: see text], where [Formula: see text] denotes the number of vertices of [Formula: see text] that have a smaller distance in [Formula: see text] to [Formula: see text] than to [Formula: see text], and analogously for [Formula: see text]. We determine all the graphs that maximize and minimize the Mostar index, respectively, over all trees in terms of some fixed parameters like the number of odd vertices, the number of vertices of degree two, and the number of pendent paths of fixed length.

On Unicyclic Graphs with Minimum Graovac–Ghorbani Index

In discrete mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. Chemical graph theory is concerned with non-trivial applications of graph theory to the solution of molecular problems. Its main goal is to use numerical invariants to reduce the topological structure of a molecule to a single number that characterizes its properties. Topological indices are numerical invariants associated with the chemical constitution, for the purpose of the correlation of chemical structures with various physical properties, chemical reactivity, or biological activity. They have found important application in predicting the behavior of chemical substances. The Graovac–Ghorbani (ABCGG) index is a topological descriptor that has improved predictive potential compared to analogous descriptors. It is used to model both the boiling point and melting point of molecules and is applied in the pharmaceutical industry. In the recent years, the number of publications on its mathematical properties has increased. The aim of this work is to partially solve an open problem, namely to find the structure of unicyclic graphs that minimize the ABCGG index. We characterize unicyclic graphs with even girth that minimize the ABCGG index, while we also present partial results for odd girths. As an auxiliary result, we compare the ABCGG indices of paths and cycles with an odd number of vertices.

Local Spin Frustration and Order in Fractional Cairo Pentagonal Lattice Under Magnetic Fields

Pentagonal lattices with an odd number of vertices are known to exhibit spin frustration, where the orientation of the spins in the ground state is not uniquely determined. In this study, we focus on a lattice created by incorporating fractals within a Cairo pentagonal lattice. We refer to this lattice as cairoflake lattice, analogous to the well-known fractal pentaflake. We use the Monte Carlo method to study an antiferromagnetic Ising model in a magnetic field for the ground state of the cairoflake lattice. Our findings reveal a separation of the spin order state and the frustration-induced disorder state in the paramagnetic phase near the magnetic-field-induced ferromagnetic phase.

Lexicographic palindromic products

A graph G on n vertices is palindromic if there is a vertex-labeling bijection f : V(G) → {1, 2, …, n} with the property that for any edge vw ∈ E(G), there is an edge xy ∈ E(G) for which f(x) = n − f(v) + 1 and f(y) = n − f(w) + 1. We characterize the conditions under which a lexicographic product G ∘ H is palindromic, as follows: If G ∘ H has an even number of vertices, then it is palindromic if and only if at least one of G or H is palindromic. If G ∘ H has an odd number of vertices, then it is palindromic if and only if both G and H are palindromic.

Independence equivalence classes of paths

The independence equivalence class of a graph G is the set of graphs that have the same independence polynomial as G. A graph whose independence equivalence class contains only itself, up to isomorphism, is independence unique. Beaton, Brown and Cameron [2] showed that paths with an odd number of vertices are independence unique and raised the problem of finding the independence equivalence class of paths with an even number of vertices. The problem is completely solved in this paper.

Perfect forests in graphs and their extensions

AbstractLet be a graph on vertices. For , a spanning forest of is called an ‐perfect forest if every tree in is an induced subgraph of and exactly vertices of have even degree (including zero). An ‐perfect forest of is proper if it has no vertices of degree zero. Scott showed that every connected graph with an even number of vertices contains a (proper) 0‐perfect forest. We prove that one can find a 0‐perfect forest with a minimum number of edges in polynomial time, but it is NP‐hard to obtain a 0‐perfect forest with a maximum number of edges. Moreover, we show that to decide whether has a 0‐perfect forest with at least edges, where is the parameter, is W[1]‐hard. We also prove that for a prescribed edge of it is NP‐hard to obtain a 0‐perfect forest containing but one can decide if there exists a 0‐perfect forest not containing in polynomial time. It is easy to see that every connected graph with an odd number of vertices has a 1‐perfect forest. It is not the case for proper 1‐perfect forests. We give a characterization of when a connected graph has a proper 1‐perfect forest.

Berry phase in the rigid rotor: Emergent physics of odd antiferromagnets

The rigid rotor is a classic problem in quantum mechanics, describing the dynamics of a rigid body with its center of mass held fixed. It is well known to describe the rotational spectra of molecules. The configuration space of this problem is SO(3), the space of all rotations in three dimensions. This is a topological space with two types of closed loops: trivial loops that can be adiabatically shrunk to a point and nontrivial loops that cannot. In the traditional formulation of the problem, stationary states are periodic over both types of closed loops. However, periodicity conditions may change if Berry phases are introduced. We argue that time-reversal symmetry allows for only one new possibility---a Berry phase of $\ensuremath{\pi}$ attached to all nontrivial loops. We derive the corresponding stationary states by exploiting the connection between SO(3) and SU(2) spaces. The solutions are antiperiodic over any nontrivial loop; i.e., stationary states reverse sign under a $2\ensuremath{\pi}$ rotation about any axis. Remarkably, this framework is realized in the low-energy physics of certain quantum magnets. The magnets must satisfy the following conditions: (a) the classical ground states are unpolarized, carrying no net magnetization, (b) the set of classical ground states is indexed by SO(3), and (c) the product $N\ifmmode\times\else\texttimes\fi{}S$ is a half-integer, where $N$ is the number of spins and $S$ is the spin quantum number. We demonstrate this result in a family of Heisenberg antiferromagnets defined on polygons with an odd number of vertices. At each vertex, we have a spin-$S$ moment that is coupled to its nearest neighbors. In the classical limit, these magnets have coplanar ground states. Their quantum spectra, at low energies, correspond to ``spherical top'' and ``symmetric top'' rigid rotors. For integer values of $S$, we recover traditional rigid-rotor spectra. With half-integer $S$, we obtain rotor spectra with a Berry phase of $\ensuremath{\pi}$.

Regular Graphs with Few Longest Cycles

Motivated by work of Haythorpe, Thomassen and the author showed that there exists a positive constant $c$ such that there is an infinite family of 4-regular 4-connected graphs, each containing exactly $c$ Hamiltonian cycles. We complement this by proving that the same conclusion holds for planar 4-regular 3-connected graphs, although it does not hold for planar 4-regular 4-connected graphs by a result of Brinkmann and Van Cleemput [European J. Combin., 97 (2021), 103395], and that it holds for 4-regular graphs of connectivity 2 with the constant $144 < c$, which we believe to be minimal among all Hamiltonian 4-regular graphs of sufficiently large order. We then disprove a conjecture of Haythorpe by showing that for every nonnegative integer $k$ there is a 5-regular graph on $26 + 6k$ vertices with $2^{k+10} \cdot 3^{k+3}$ Hamiltonian cycles. We prove that for every $d \ge 3$ there is an infinite family of Hamiltonian 3-connected graphs with minimum degree $d$, with a bounded number of Hamiltonian cycles. It is shown that if a 3-regular graph $G$ has a unique longest cycle $C$, at least two components of $G - E(C)$ have an odd number of vertices on $C$, and that there exist 3-regular graphs with exactly two such components.

Infinite chromatic games

In the paper we introduce a new variant of the graph coloring game and a new graph parameter being the result of the new game. We study their properties and get some lower and upper bounds, exact values for complete multipartite graphs and optimal, often polynomial-time strategies for both players provided that the game is played on a graph with an odd number of vertices. At the end we show that both games, the new and the classic one, are related: our new parameter is an upper bound for the game chromatic number.

Filling in pattern designs for incomplete pairwise comparison matrices: (Quasi-)regular graphs with minimal diameter

Pairwise comparisons have become popular in the theory and practice of preference modelling and quantification. In case of incomplete data, the arrangements of known comparisons are crucial for the quality of results. We focus on decision problems where the set of pairwise comparisons can be chosen and it is designed completely before the decision making process, without any further prior information. The objective of this paper is to provide recommendations for filling patterns of incomplete pairwise comparison matrices based on their graph representation. The proposed graphs are regular and quasi-regular ones with minimal diameter (longest shortest path). Regularity means that each item is compared to others for the same number of times, resulting in a kind of symmetry. A graph on an odd number of vertices is called quasi-regular, if the degree of every vertex is the same odd number, except for one vertex whose degree is larger by one. We draw attention to the diameter, which is missing from the relevant literature, in order to remain the closest to direct comparisons. If the diameter of the graph of comparisons is as low as possible (among the graphs of the same number of edges), we can decrease the cumulated errors that are caused by the intermediate comparisons of a long path between two items. Contributions of this paper include a list containing (quasi-)regular graphs with diameter 2 and 3 up until 24 vertices. Extensive numerical tests show that the recommended graphs indeed lead to better weight vectors compared to various other graphs with the same number of edges. It is also revealed by examples that neither regularity nor small diameter is sufficient on its own, both properties are needed. Both theorists and practitioners can utilize the results, given in several formats in the appendix: plotted graph, adjacency matrix, list of edges, ‘Graph6’ code.

A Combinatorial Approach to the Computation of the Fractional Edge Dimension of Graphs

E. Yi recently introduced the fractional edge dimension of graphs. It has many applications in different areas of computer science such as in sensor networking, intelligent systems, optimization, and robot navigation. In this paper, the fractional edge dimension of vertex and edge transitive graphs is calculated. The class of hypercube graph Qn with an odd number of vertices n is discussed. We propose the combinatorial criterion for the calculation of the fractional edge dimension of a graph, and hence applied it to calculate the fractional edge dimension of the friendship graph Fk and the class of circulant graph Cn(1,2).

Turán numbers for disjoint paths

AbstractThe Turán number of a graph , ex, is the maximum number of edges in a graph of order that does not contain a copy of as a subgraph. Lidický, Liu and Palmer determined ex for sufficiently large and proved that the extremal graph is unique, where is the union of pairwise vertex‐disjoint paths on vertices. There are a few kinds of graphs for which ex are known exactly for all , including cliques, matchings, paths, cycles on odd number of vertices and some other special graphs. In this paper, using a different approach from Lidický, Liu and Palmer, we determine ex for all integers when at most one of is odd. Furthermore, we show that there exists a family of pairs of bipartite graphs such that for all integers , which is related to an old problem of Erdős and Simonovits.

The Maximal 1-Planarity and Crossing Numbers of Graphs

A 1-planar graph is a graph which has a drawing on the plane such that each edge has at most one crossing. Czap and Hudak showed that every 1-planar graph with n vertices has crossing number at most $$n-2$$ . In this paper, we prove that every maximal 1-planar graph G with n vertices has crossing number at most $$n-2-(2\lambda _1+2\lambda _2+\lambda _3)/6$$ , where $$\lambda _1$$ and $$\lambda _2$$ are respectively the numbers of 2-degree and 4-degree vertices in G, and $$\lambda _3$$ is the number of odd vertices w in G such that either $$d_G(w)\le 9$$ or $$G-w$$ is 2-connected. Furthermore, we show that every 3-connected maximal 1-planar graph with n vertices and m edges has crossing number $$m-3n+6$$ .

Tetravalent edge-transitive Cayley graphs of Frobenius groups

In this paper, we give a characterisation for a class of edge-transitive Cayley graphs and provide a method for constructing edge-transitive graphs of valency 4 with arbitrarily large vertex stabiliser. In particular, in the last section, we obtain certain extensions of the results of Li et al. (Tetravalent edge-transitive Cayley graphs with odd number of vertices, J Comb Theory Ser B 96:164–181, 2006) on half-transitive graphs.

HAMILTONICITY OF CAMOUFLAGE GRAPHS

This paper examines the Hamiltonicity of graphs having some hidden behaviours of some other graphs in it. The well-known mathematician Barnette introduced the open conjecture which becomes a theorem by restricting our attention to the class of graphs which is 3-regular, 3- connected, bipartite, planar graphs having odd number of vertices in its partition be proved as a Hamiltonian. Consequently the result proved in this paper stated that “Every connected vertex-transitive simple graph has a Hamilton path” shows a significant improvement over the previous efforts by L.Babai and L.Lovasz who put forth this conjecture. And we characterize a graphic sequence which is forcibly Hamiltonian if every simple graph with degree sequence is Hamiltonian. Thus we discussed about the concealed graphs which are proven to be Hamiltonian.

Minimum status of trees with given parameters

The status of a vertex v in a connected graph G is defined as the sum of the distances from v to all other vertices in G. The minimum status of G is the minimum of status of all vertices of G. We give the smallest and largest values for the minimum status of a tree with fixed parameters such as the diameter, the number of pendant vertices, the number of odd vertices, and the number of vertices of degree two, and characterize the unique extremal trees.

Minimal Non-Odd-Transversal Hypergraphs and Minimal Non-Odd-Bipartite Hypergraphs

Among all uniform hypergraphs with even uniformity, the odd-transversal or odd-bipartite hypergraphs are closer to bipartite simple graphs than bipartite hypergraphs from the viewpoint of both structure and spectrum. A hypergraph is called odd-transversal if it contains a subset of the vertex set such that each edge intersects the subset in an odd number of vertices, and it is called minimal non-odd-transversal if it is not odd-transversal but deleting any edge results in an odd-transversal hypergraph. In this paper we give an equivalent characterization of the minimal non-odd-transversal hypergraphs by means of the degrees and the rank of its incidence matrix over $\mathbb{Z}_2$. If a minimal non-odd-transversal hypergraph is uniform, then it has even uniformity, and hence is minimal non-odd-bipartite. We characterize $2$-regular uniform minimal non-odd-bipartite hypergraphs, and give some examples of $d$-regular uniform hypergraphs which are minimal non-odd-bipartite. Finally we give upper bounds for the least H-eigenvalue of the adjacency tensor of minimal non-odd-bipartite hypergraphs.

On distance matrices of wheel graphs with an odd number of vertices

Let denote the wheel graph having n-vertices. If i and j are any two vertices of , define Let D be the matrix with entry equal to . The matrix D is called the distance matrix of . Suppose is an odd integer. In this paper, we deduce a formula to compute the Moore-Penrose inverse of D. More precisely, we obtain an matrix and a rank one matrix such that Here, is positive semidefinite, and all row sums are equal to zero.