Empty Graph Research Articles

Counting spanning trees of multiple complete split-like graph containing a given spanning forest

The multiple complete split-like graph MCSb,sa is the join of an empty graph K‾a and s copies of complete graph Kb. In this article, we obtain the formulas for the number of spanning trees of MCSb,sa containing a given spanning forest when s=1 and 2. Particularly, when s=1, our result derives the number of spanning trees of complete split graph containing a given spanning forest, thereby extending Moon's result [19].

EDGE IRREGULAR REFLEXIVE LABELING OF DUMBBELL GRAPH, CORONA OF OPEN LADDER, AND NULL GRAPH

Graph is a simple, connected, undirected graph with vertex set and edge set . A graph is called to have an edge irregular reflexive -labeling if its vertices can be labeled with even numbers from until and its edges can be labeled with positive integers from to such that the weights for all the edges are different, where . The weight of edge uv in graph with labeling, denoted by , is defined as sum of the edge label and all vertex labels incident to that edge. The reflexive edge strength of a graph , denoted by , is the value of minimum of the largest label. In this paper, edge irregular reflexive -labeling for Dumbbell Graph and corona of open ladder and null graph will be determined. The reflexive edge strength of the Dumbbell Graph with and is for and for The reflexive edge strength of the corona of open ladder and null graph with n ≥ 3 and m ≥ 1 is for and for .

THE REFLEXIVE EDGE STRENGTH OF THE PENTAGONAL SNAKE GRAPH AND CORONA OF THE OPEN TRIANGULAR LADDER AND NULL GRAPH

Assume that be an undirected simple graph with vertex set and edge set . The edge irregular reflexive -labeling of graph is a labeling selects positive integers from 1 to as edge labels and non negative even numbers from 0 to as vertex labels, and the weights assigned to each edge are distinct, where . On graph with labeling, the weight of edge is represented by which is defined as the sum of edge label and all vertex labels incident to that edge. Reflexive edge strength of graph is the minimum of the highest label, denoted by . In this research, reflexive edge strength for pentagonal snake graph and corona of open triangular ladder and null graph will be determined. The method of this research is literature study, the lower bound of determined by Ryan’s lemma and the upper bound by labeling. The reflexive edge strength of pentagonal snake graph with is for and for The reflexive edge strength of corona of open triangular ladder and null graph with n ≥ 3 and m ≥ 1 is and .

Asymptotic dynamic graph order evolution analysis

In this work, we investigate the analysis of generators for dynamic graphs, which are defined as graphs whose topology changes over time. We focus on generated graphs whose order (number of nodes) varies over time. We use a concept called “sustainability” to qualify the long-term evolution of dynamic graphs. A dynamic graph is considered sustainable if its evolution does not result in a static, empty, or periodic graph. To illustrate how the analysis can be conducted, a parameterized and probability-based generator, named D3G3 (Degree-Driven Dynamic Geometric Graphs Generator), has been introduced in a recent work. From this model, we derive multiple scenarios that correspond to three trends in graph order evolution. Our central contribution lies in a mathematical framework that provides an expectation of the order of the graph at time step t+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t+1$$\\end{document}, given its order at time step t. Nevertheless, our analysis underscores the challenge of characterizing the sustainability of dynamic graphs, even when a formal mathematical model for graph order evolution is known.

Catalogue of Sceliphron destillatorium (Illager, 1807) (Hymenoptera, Sphecidae) specimens deposited in the State Museum of Natural History NASU, Lviv, Ukraine

Representatives of the genus Sceliphron Klug, 1801 are found in almost all biogeographical regions of the world (Yuan, 2022). At present, 35 species of this genus have been known (Pulawski, 2020). Of these, six species have been recorded from the territory of Ukraine as follows: three invasive species: Sceliphron caementarium (Drury, 1773); Sceliphron curvatum (Smith, 1870) and Sceliphron deforme (F. Smith, 1856) (Kletʹonkin, 2023); and three native species: Sceliphron destillatorium (Illiger, 1807); Sceliphron madraspatanum (Fabricius, 1781) and Sceliphron spirifex (Linnaeus, 1758) (Tymkiv et al., 2015). The most common is Sceliphron destillatorium (Illiger, 1807), which occurs throughout the territory of Ukraine. Representatives of the genus Sceliphron Klug, 1801 belong to two subgenera: Sceliphron and Hensenia. Although species of both subgenera take care of their offspring and feed the larvae with spiders, their nests differ in shape and number of mud cells. Wasps of the subgenus Hensenia build a certain number of single-celled mud nests in shape of jugs. Instead, wasps of the subgenus Sceliphron build cells that joint into large mud nest. S. destillatorium is the South–Palaearctic species. Its range covers the Mediterranean region (southern and south-eastern Europe, northern Africa and south-western and southern Asia) (Wisniowski et al., 2013). S. destillatorium is a solitary wasp. It belongs to the family Sphecidae (Insecta: Hymenoptera) of the subgenus Sceliphron (Gülmez1 et al, 2017). S. destillatorium female builds a nests of the moist soil material, which she collects and transports herself. The nest consists of tubular cells placed side by side, the number of which varies (Chatenoud et al., 2012). After building the first cell, the female hunts spiders to feed her offspring (Pytel-Huta, 2023; Yuan, 2022). Female S. destillatorium brings the paralysed spider into the cell, lays an egg and continues to hunt. Each cell stores a certain number of spiders. The number depends on the size of the victims and the sex of the future generation. Once the cells have been built, the female S. destillatorium covers them with an additional layer of soil (Polidori, 2005). In the wild, females build their nests in shady, sheltered places (rock ledges, empty tree trunks, etc.). However, at present, they are increasingly nesting in people's houses (usually under roofs, attics, etc.) (Yuan, 2022). Adult wasps feed on angiosperm nectar. The collection of the State Museum of Natural History, National Academy of Sciences in Lviv contains 14 specimens of this species collected in the western part of Ukraine and one specimen collected in the central part. A few samples (4) of this species were collected in 1922, 1923 and 1926, collector – J. Noskiewicz. The date and place of collection of the remaining specimens are unknown. All data have been entered into the web resource Data Centre “Biodiversity of Ukraine” http://dc.smnh.org/.

Behaviour of the minimum degree throughout the ${\textit{d}}$ -process

AbstractThe $d$ -process generates a graph at random by starting with an empty graph with $n$ vertices, then adding edges one at a time uniformly at random among all pairs of vertices which have degrees at most $d-1$ and are not mutually joined. We show that, in the evolution of a random graph with $n$ vertices under the $d$ -process with $d$ fixed, with high probability, for each $j \in \{0,1,\dots,d-2\}$ , the minimum degree jumps from $j$ to $j+1$ when the number of steps left is on the order of $\ln (n)^{d-j-1}$ . This answers a question of Ruciński and Wormald. More specifically, we show that, when the last vertex of degree $j$ disappears, the number of steps left divided by $\ln (n)^{d-j-1}$ converges in distribution to the exponential random variable of mean $\frac{j!}{2(d-1)!}$ ; furthermore, these $d-1$ distributions are independent.

Product of Semi – Lattices of Certain Graphs

Introduction: In this article, author tries to construct a relation between graphs of product of meet-semilattices L = L_1 X L_2, where L_1 and L_2 are two semilattices and obtain some properties of such graphs. Author investigated that for meet-semilattices L_1 and L_2 has a cycle of length n-1 and n.
 Objectives: author reveals that if L_1 and L_2 be two meet-semilattices with 0 and L = L_1 X L_2, then it is a star graph. In this paper, we have covered some definitions, examples and theorems on zero devisor graph edge of a 4 - cycles or a 5 - cycles. Γ(L) is a star graph.
 Methods: Theorem 1.1 [9] The zero-divisor graph of a finite meet-semilattice with only one atom is the empty graph. The zero-divisor graph of the meet semilattice is the empty graph. However, this does not hold for infinite meet-semilattices with one atom. For consider, the infinite meet-semilattice, where the descending dots represent infinite descending chain. It has only one atom c but its graph Γ(L) is an infinite star graph.
 Theorem 1.2 [12] Every disconnected graph cannot be a graph of any meet- semilattice L with 0.
 Remark 1.2 A graphs of product of meet-semilattices and obtain some properties of such graphs. In this section, we consider two integral meet-semilattices L_1 and L_2 with L ∼=L_1 X L_2 and show that if ∣L_1∣ = m+1, ∣L_2∣ = n+1, then Γ(L) is the complete bipartite graph K (m, n).
 Results: If L does not contain any atom, then any edge in Γ(L) is contained in a cycle of length ≤ 6, and therefore Γ(L) is a union of 4 - cycles and 5 - cycles. Let L be a meet-semilattice with 0. If Γ(L) contains a cycle, then the core K of Γ(L) is a union of 4- cycles and 5 – cycles and any vertex in Γ(L) is either a vertex of the core K of Γ(L) or is a pendant of Γ(L). Let L_1 and L_2 be two meet-semilattices with 0 and L = L_1 X L_2. Then exactly one of the following holds:
 
 Γ(L) has a cycle of length n-1 or n (that is gr Γ(L)≤ n).
 Γ(L) is a star graph.
 
 Conclusions: In this article we have studied the concept of the zero-divisor graph derived from meet-semilattice L with 0 on the lines of Anderson and Livingston [6]. Also, we generalized certain results from Demeyer, Mckenzie and Schneider [18] to meet-semilattice L with 0.

On the Enhanced Power Graph of a Semigroup

The enhanced power graph [Formula: see text] of a semigroup [Formula: see text] is a simple graph whose vertex set is [Formula: see text] and two vertices [Formula: see text], [Formula: see text] are adjacent if and only if [Formula: see text], [Formula: see text] for some [Formula: see text], where [Formula: see text] is the subsemigroup generated by [Formula: see text]. In this paper, we first describe the structure of [Formula: see text] for an arbitrary semigroup [Formula: see text], and then discuss the connectedness of [Formula: see text]. Further, we characterize the semigroup [Formula: see text] in the cases when [Formula: see text] is separately a complete, bipartite, regular, tree and null graph. The planarity, together with the minimum degree and independence number, of [Formula: see text] is also investigated. The chromatic number of a spanning subgraph, i.e., the cyclic graph, of [Formula: see text] is proved to be countable. In the final part of this paper, we construct an example of a semigroup [Formula: see text] such that the chromatic number of [Formula: see text] need not be countable.

Co-eigenvector graphs

Except for the empty graph, we show that the orthogonal matrix X of the adjacency matrix A determines that adjacency matrix completely, but not always uniquely. The proof relies on interesting properties of the Hadamard product Ξ=X∘X. As a consequence of the theory, we show that irregular co-eigenvector graphs exist only if the number of nodes N≥6. Co-eigenvector graphs possess the same orthogonal eigenvector matrix X, but different eigenvalues of the adjacency matrix. Co-eigenvector graphs are the dual of co-spectral graphs, that share all eigenvalues of the adjacency matrix, but possess a different orthogonal eigenvector matrix. We deduce general properties of co-eigenvector graph and start to enumerate all co-eigenvector graphs on N=6 and N=7 nodes. Finally, we list many open problems.

Power of $k$ Choices in the Semi-Random Graph Process

The semi-random graph process is a single player game in which the player is initially presented an empty graph on $n$ vertices. In each round, a vertex $u$ is presented to the player independently and uniformly at random. The player then adaptively selects a vertex $v$, and adds the edge $uv$ to the graph. For a fixed monotone graph property, the objective of the player is to force the graph to satisfy this property with high probability in as few rounds as possible.
 In this paper, we introduce a natural generalization of this game in which $k$ random vertices $u_1, \ldots, u_k$ are presented to the player in each round. She needs to select one of the presented vertices and connect to any vertex she wants. We focus on the following three monotone properties: minimum degree at least $\ell$, the existence of a perfect matching, and the existence of a Hamiltonian cycle.

Sharp thresholds in adaptive random graph processes

AbstractThe ‐process is a single player game in which the player is initially presented the empty graph on vertices. In each step, a subset of edges is independently sampled according to a distribution . The player then selects one edge from , and adds to its current graph. For a fixed monotone increasing graph property , the objective of the player is to force the graph to satisfy in as few steps as possible. The ‐process generalizes both the Achlioptas process and the semi‐random graph process. We prove a sufficient condition for the existence of a sharp threshold for in the ‐process. Using this condition, in the semi‐random process we prove the existence of a sharp threshold when corresponds to being Hamiltonian or to containing a perfect matching. This resolves two of the open questions proposed by Ben‐Eliezer et al. (RSA, 2020).

On the j-Edge Intersection Graph of Cycle Graph

This paper defines a new class of graphs using the spanning subgraphs of a cycle graph as vertices. This class of graphs is called $j$-edge intersection graph of cycle graph, denoted by $E_{C_{(n,j)}}$. The vertex set of $E_{C_{(n,j)}}$ is the set of spanning subgraphs of cycle graph with $j$ edges where $n \geq 3$ and $j$ is a nonnegative integer such that $1 \leq j \leq n$. Moreover, two distinct vertices are adjacent if they have exactly one edge in common. $E_{C_{(n,j)}}$ is considered as a simple graph. Furthermore, $E_{C_{(n,j)}}$ is characterized by the value of $j$ that is when $j=1$ or $\lceil \frac{n}{2} \rceil < j \leq n$ and $2 \leq j \leq \lceil \frac{n}{2} \rceil$. When $j=1$ or $\lceil \frac{n}{2} \rceil < j \leq n$, the new graph only produced an empty graph. Hence, the proponents only considered the value when $2 \leq j \leq \lceil \frac{n}{2} \rceil$ in determining the order and size of $E_{C_{(n,j)}}$. Moreover, this paper discusses necessary and sufficient conditions where the $j$-edge intersection graph of $C_n$ is isomorphic to the cycle graph. Furthermore, the researchers determined a lower bound for the independence number, and an upper bound for the domination number of $E_{C_{(n,j)}}$ when $j=2$.

On local antimagic chromatic numbers of circulant graphs join with null graphs or cycles

An edge labeling of a graph G = (V,E) is said to be local antimagic if there is a bijection f : E → {1,..., |E|} such that for any pair of adjacent vertices x and y, f +(x) ≠ f +(y), where the induced vertex label is f +(x) = 𝜮 f(e), with e ranging over all the edges incident to x. The local antimagic chromatic number of G, denoted by χla(G), is the minimum number of distinct induced vertex labels over all local antimagic labelings of G. For a bipartite circulant graph G, it is known that χ(G)=2 but χla(G) ≥ 3. Moreover, χla(Cn ∨ K1)=3 (respectively 4) if n is even (respectively odd). Let G be a graph of order m ≥ 3. In [Affirmative solutions on local antimagic chromatic number, Graphs Combin., 36 (2020), 1337—1354], the authors proved that if m ≡ n (mod 2) with χla(G) = χ(G), m>n ≥ 4 and m ≥ n2/2, then χla(G∨On) = χla(G)+1. In this paper, we show that the conditions can be omitted in obtaining χla(G∨H) for some circulant graph G, and H is a null graph or a cycle. The local antimagic chromatic number of certain wheel related graphs are also obtained.

Graphs of fixed order and size with maximal Aα-index

For any real number α∈[0,1], by the Aα-matrix of a graph G we mean the matrix Aα(G)=αD(G)+(1−α)A(G), where A(G) and D(G) are the adjacency matrix and the diagonal matrix of vertex degrees of G, respectively. The largest eigenvalue of Aα(G) is called the Aα-index of G. In this paper, we settle the problem of characterizing graphs which attain the maximum Aα-index over G(n,n+k), the class of graphs with n vertices and n+k edges, for −1≤k≤n−3 and 12≤α<1. The following result is obtained: for −1≤k≤n−3, when 12≤α<1, Hn,k is the unique graph in G(n,n+k) that maximizes the Aα-index, except when (n,k)=(4,−1),(n,2) or (7,3) and α=12, or (n,k)=(5,1) and α∈[12,35−40924]. When (n,k,α)=(4,−1,12), the optimal graphs are H4,−1 and K3∪K1; when (n,k,α)=(n,2,12), the optimal graphs are Hn,2 and Gn,2; when (n,k,α)=(5,1,35−40924), the optimal graphs are H5,1 and K4∪K1; when (n,k,α)=(7,3,12), the optimal graphs are H7,3 and K5∪2K1; when (n,k)=(5,1) and 12≤α<35−40924, K4∪K1 is the unique graph that maximizes the Aα-index. Our work completes the corresponding work of Chang and Tam (2010) and Zhai et al. (2022) for the special case α=12. As a by-product, we provide a new proof for the known result that for any positive integer m and any real number α∈[12,1), if (m,α)≠(3,12), then a graph maximizes the Aα-index over all graphs with m edges if and only if it is the union of K1,m with a (possibly empty) null graph; a graph maximizes the A12-index over all graphs with three edges if and only if it is the union of K1,3 or K3 with a (possibly empty) null graph. Some open questions are also posed.

Spanning graph topology of graphs

The collection of subgraphs of a graph [Formula: see text] containing [Formula: see text] and the null graph [Formula: see text], which is closed under union and intersection, is said to be a graph topology defined on [Formula: see text]. In this paper, we investigate the idea of spanning graph topology of a graph [Formula: see text], where we consider the collection of spanning subgraphs of [Formula: see text] satisfying the axioms analogous to the axioms of graph topology. We begin with the basic concepts of spanning graph topological space and later and introduce two spanning subgraph complements to define closed graphs in spanning graph topological spaces.

The Early Evolution of the Random Graph Process in Planar Graphs and Related Classes

We study the random planar graph process introduced by Gerke et al. [Random Structures Algorithms, 32 (2008), pp. 236–261]: Begin with an empty graph on vertices, consider the edges of the complete graph one by one in a random ordering, and at each step add an edge to a current graph only if the graph remains planar. They studied the number of edges added up to step for “large" . In this paper we extend their results by determining the asymptotic number of edges added up to step in the early evolution of the process when . We also show that this result holds for a much more general class of graphs, including outerplanar graphs, planar graphs, and graphs on surfaces.

Semi-random process without replacement

Semi-random processes involve an adaptive decision-maker, whose goal is to achieve some pre-determined objective in an online randomized environment. We introduce and study a semi-random multigraph process, which forms a no-replacement variant of the process that was introduced by Ben-Eliezer, Hefetz, Kronenberg, Parczyk, Shikhelman and Stojakovi\'c (2020). The process starts with an empty graph on the vertex set $[n]$. For every positive integers $q$ and $1\leq r\leq n$, in the $((q-1)n+r)$th round of the process, the decision-maker, called \emph{Builder}, is offered the vertex $\pi_q(r)$, where $\pi_1, \pi_2, \ldots$ is a sequence of permutations in $S_n$, chosen independently and uniformly at random. Builder then chooses an additional vertex (according to a strategy of his choice) and connects it by an edge to $\pi_q(r)$. For several natural graph properties, such as $k$-connectivity, minimum degree at least $k$, and building a given spanning graph (labeled or unlabeled), we determine the typical number of rounds Builder needs in order to construct a graph having the desired property. Along the way we introduce and analyze an urn model which may also have independent interest.

Partially balanced incomplete block (PBIB)-designs arising from diametral paths in some strongly regular graphs

The partially balanced incomplete block (PBIB)-designs and association schemes arising from different graph parameters is a well-studied concept. In this paper, we define and construct PBIB-designs with association schemes in strongly regular graphs through the nodes belonging to the diametral paths as blocks. A [Formula: see text]-design, called a diametral-design (in short) over a strongly regular graph [Formula: see text] of degree [Formula: see text], diameter [Formula: see text], is an ordered pair [Formula: see text], where [Formula: see text] and [Formula: see text] is the set of all diametral paths of [Formula: see text], called blocks, containing the nodes belonging to the diametral paths, satisfying the following conditions: (i) If [Formula: see text] and [Formula: see text], then there are exactly [Formula: see text] blocks containing [Formula: see text]; (ii) If [Formula: see text], [Formula: see text] and [Formula: see text], then there are exactly [Formula: see text]-blocks containing [Formula: see text]. We construct PBIB-designs with two-association schemes through diametral paths corresponding to the strongly regular graphs such as the square lattice graphs [Formula: see text], the triangular graphs [Formula: see text], the three Chang graphs, the Petersen graph, the Shrikhande graph, the Clebsch graph, the complement of Clebsch graph, the complement of Schläfli graph, complete bipartite graphs, the Hoffman–Singleton graph, etc. and in each case we give the composition of the diametral paths. These serve as extensions for the collection of near impossible class of strongly regular graphs with given parameters having two-association schemes.

Online and Connected Online Ramsey Numbers of a Matching versus a Path

The (G1,G2)-online Ramsey game is a two-player turn-based game between a builder and a painter. Starting from an empty graph with infinite vertices, the builder adds a new edge in each round, and the painter colors it red or blue. The builder aims to force either a red copy of G1 or a blue copy of G2 in as few rounds as possible, while the painter’s aim is the opposite. The online Ramsey number r˜(G1,G2) is the minimum number of edges that the builder needs to win the (G1,G2)-online Ramsey game, regardless of the painter’s strategy. Furthermore, we initiate the study of connected online Ramsey game, which is identical to the usual one, except that at any time the graph induced by all edges should be connected. In this paper, we show a general bound of the online Ramsey number of a matching versus a path and determine its exact value when the path has an order of three or four. For the connected version, we obtain all connected online Ramsey numbers of a matching versus a path.

Majorana algebra for the Hoffman–Singleton graph

Majorana theory is an axiomatic tool for studying the Monster group M and its subgroups through the 196,884-dimensional Conway–Griess–Norton algebra. The theory was introduced by A. A. Ivanov in 2009 and since then it experienced a remarkable development including the classification of Majorana representations for small (and not so small) groups. The group U_3(5) is (isomorphic to) the socle of the centralizer in M of a subgroup of order 25. The involutions of this U_3(5)-subgroup are 2A-involutions in the Monster. Therefore, U_3(5) possesses a Majorana representation (based on the embedding in the Monster). We prove that this is the unique Majorana representation of U_3(5), calculate its dimension, which is 798, and obtain a description in terms of the Hoffman–Singleton graph of which the automorphism group has U_3(5) as an index 2 subgroup.