Adjacent Vertices Research Articles

Distance Pattern Distinguishing Coloring of Graphs

Given a connected (p, q)− graph G = (V, E) of diameter d, ∅M ⊆ V (G) and a nonempty set X = {0, 1, ..., d} of colors of cardinality , let fM be an assignment of subsets of X to the vertices of G such that fM(u) = {d(u, v) : v ∈ M} where, d(u, v) is the usual distance between u and v . We call fM an M− distance pattern coloring of G if no two adjacent vertices have same fM. Define f ⊕ M of an edge e ∈ E(G) as f ⊕ M(e) = fM(u) ⊕ fM(v); e = uv. A distance pattern distinguishing coloring of a graph G is an M distance pattern coloring of G such that both fM(G) and f ⊕ M(G) are injective. This paper is a study on distance pattern coloring and distance pattern distinguishing coloring of graphs.

Infinite constant gap length trees in products of thick Cantor sets

We show that products of sufficiently thick Cantor sets generate trees in the plane with constant distance between adjacent vertices. Moreover, we prove that the set of choices for this distance has non-empty interior. We allow our trees to be countably infinite, which further distinguishes this work from previous results on patterns in fractal sets. This builds on the authors’ previous work on graphs and distance sets over products of Cantor sets of sufficient Newhouse thickness.

Fault-tolerant strong Menger connectivity of modified bubble-sort graphs

An interconnection network is usually modeled as a graph, in which vertices and edges correspond to processors and communication links, respectively. For the study of graphs, connectivity and edge connectivity are the basic issues, and the discussion of fault tolerance is based on connectivity. A connected graph G is strongly Menger connected if each pair of vertices u and v of G, there are min{dG(u),dG(v)} vertex-disjoint paths connecting u and v. G is m-fault-tolerant strongly Menger connected if G−F remains strongly Menger connected for any F⊆V(G) with |F|≤m. G is m-conditional fault-tolerant strongly Menger connected if G−F remains strongly Menger connected for any F⊆V(G) with |F|≤m and δ(G−F)≥2. In this paper, we demonstrate that n-dimensional modified bubble-sort graphs MBn is (n−2)-fault-tolerant strongly Menger connected and (n−2)-fault-tolerant one-to-many strongly Menger connected for n≥4. Moreover, under the restricted condition that each vertex has at least two fault-free adjacent vertices, MBn is (2n−5)-conditional fault-tolerant strongly Menger connected for n≥4.

Super-Connectivity of the Folded Locally Twisted Cube

The hypercube Qn is one of the most popular interconnection networks with high symmetry. To reduce the diameter of Qn, many variants of Qn have been proposed, such as the n-dimensional locally twisted cube LTQn. To further optimize the diameter of LTQn, the n-dimensional folded locally twisted cube FLTQn is proposed, which is built based on LTQn by adding 2n−1 complementary edges. Connectivity is an important indicator to measure the fault tolerance and reliability of a network. However, the connectivity has an obvious shortcoming, in that it assumes all the adjacent vertices of a vertex will fail at the same time. Super-connectivity is a more refined index to judge the fault tolerance of a network, which ensures that each vertex has at least one neighbor. In this paper, we show that the super-connectivity κ(1)(FLTQn)=2n for any integer n≥6, which is about twice κ(FLTQn).

AGMN: Association graph-based graph matching network for coronary artery semantic labeling on invasive coronary angiograms

Semantic labeling of coronary arterial segments in invasive coronary angiography (ICA) is important for automated assessment and report generation of coronary artery stenosis in computer-aided coronary artery disease (CAD) diagnosis. However, separating and identifying individual coronary arterial segments is challenging because morphological similarities of different branches on the coronary arterial tree and human-to-human variabilities exist. Inspired by the training procedure of interventional cardiologists for interpreting the structure of coronary arteries, we propose an association graph-based graph matching network (AGMN) for coronary arterial semantic labeling. We first extract the vascular tree from invasive coronary angiography (ICA) and convert it into multiple individual graphs. Then, an association graph is constructed from two individual graphs where each vertex represents the relationship between two arterial segments. Thus, we convert the arterial segment labeling task into a vertex classification task; ultimately, the semantic artery labeling becomes equivalent to identifying the artery-to-artery correspondence on graphs. More specifically, the AGMN extracts the vertex features by the embedding module using the association graph, aggregates the features from adjacent vertices and edges by graph convolution network, and decodes the features to generate the semantic mappings between arteries. By learning the mapping of arterial branches between two individual graphs, the unlabeled arterial segments are classified by the labeled segments to achieve semantic labeling. A dataset containing 263 ICAs was employed to train and validate the proposed model, and a five-fold cross-validation scheme was performed. Our AGMN model achieved an average accuracy of 0.8264, an average precision of 0.8276, an average recall of 0.8264, and an average F1-score of 0.8262, which significantly outperformed existing coronary artery semantic labeling methods. In conclusion, we have developed and validated a new algorithm with high accuracy, interpretability, and robustness for coronary artery semantic labeling on ICAs.

Locally irregular edge-coloring of subcubic graphs

A graph is locally irregular if no two adjacent vertices have the same degree. A locally irregular edge-coloring of a graph G is such an (improper) edge-coloring that the edges of any fixed color induce a locally irregular subgraph. Among the graphs admitting a locally irregular edge-coloring, i.e., decomposable graphs, only one is known to require 4 colors, while for all the others it is believed that 3 colors suffice. In this paper, we prove that decomposable claw-free graphs with maximum degree 3, all cycle permutation graphs, and all generalized Petersen graphs admit a locally irregular edge-coloring with at most 3 colors. We also discuss when 2 colors suffice for a locally irregular edge-coloring of cubic graphs and present an infinite family of cubic graphs of girth 4 which require 3 colors.

Signed graphs from proper coloring of graphs

Let χ(G) denote the chromatic number of a graph G. Under the proper coloring of a graph G with χ(G) colors, we define a signed graph from it. The obtained signed graph is defined as parity colored signed graph and denoted as Sc. The signs of edges of G are defined from the colors of the vertices as +( − ) if the colors on the adjacent vertices are of the same (opposite) parity. In this paper, we initiate a study on Sc. We further investigate the chromatic rna number of some classes of graphs concerning proper coloring.

BILANGAN KROMATIK EQUITABLE PADA GRAF BINTANG, GRAF LOLIPOP, DAN GRAF PERSAHABATAN

Let G be a connected and undirected graph. Vertex coloring in a graph G is a mapping from the set of vertices in G to the set of colors such that every two adjacent vertices have different colors. There are many types of vertex coloring, such as complete coloring, k-differential coloring, and equitable coloring. Equitable coloring of G is a vertex coloring of G that satisfies the condition that for each induced color class it has an equitable cardinality with difference 0 or 1. The minimum number of colors used for such coloring of G is called the equitable chromatic number of G, denoted by χe(G). In this study, we only concern with graphs that have a central vertex, which means a vertex that is adjacent to every other vertex, in particular on the star graph (Sn), lollipop graph (Ln), and friendship graph (fn). This research aims to formulate the equitable chromatic number of the star graph (Sn), lollipop graph (Ln), and friendship graph (fn). The first step taken in this research is to apply vertex coloring to Sn, Ln, and fn. After that, the color classes of the vertex set are obtained and its cardinality is determined. Next, analyze that the applied vertex coloring meets the definition of equitable coloring. Then, prove that the number of colors used is minimum. Thus, the chromatic number for each graph is obtained and proved. Based on this research, the equitable chromatic number of Sn is ⌈n/2⌉ + 1, the equitable chromatic number of Ln is n, and the equitable chromatic number of fn is 3, for n = 1 and n + 1, for n ≥ 2.

An Algorithm for the Numbers of Homomorphisms from Paths to Rectangular Grid Graphs

Let G and H be graphs. A mapping f from the vertices of G to the vertices of H is known as a homomorphism from G to H if, for every pair of adjacent vertices x and y in G, the vertices f(x) and f(y) are adjacent in H. A rectangular grid graph is the Cartesian product of two path graphs. In this paper, we provide a formula to determine the number of homomorphisms from paths to rectangular grid graphs. This formula gives the solution to the problem concerning the number of walks in the rectangular grid graphs.

Monophonic pebbling number of some families of cycles

Let [Formula: see text] be a simple connected graph. A configuration of pebbles is a function from [Formula: see text] to a set of integers. Consider a configuration of pebbles on [Formula: see text]. A pebbling move consists of removing two pebbles off a vertex and placing one on an adjacent vertex. The monophonic pebbling number of [Formula: see text] is the smallest integer [Formula: see text] from which we can put one pebble to a target using monophonic path through pebbling moves. The monophonic [Formula: see text]-pebbling number of [Formula: see text] is the smallest positive integer [Formula: see text] such that from any configuration of [Formula: see text] pebbles we can put [Formula: see text] pebbles on a target using monophonic path. Here we discuss these concepts for families of cycles.

Klein cordial trees and odd cyclic cordial friendship graphs

For a graph G and an abelian group A, a labeling of the vertices of G induces a labeling of the edges via the sum of adjacent vertex labels. Hovey introduced the notion of an A-cordial vertex labeling when both the vertex and edge labels are as evenly distributed as possible. Much work has since been done with trees, hypertrees, paths, cycles, ladders, prisms, hypercubes, and bipartite graphs. In this paper we show that all trees are Z22-cordial except for P4 and P5. In addition, we give numerous results relating to Zm-cordiality of the friendship graph Fn. The most general result shows that when m is an odd multiple of 3, then Fn is Zm-cordial for all n. We also give a general conjecture to determine when Fn is Zm-cordial.

Solving independent set problems with photonic quantum circuits

An independent set (IS) is a set of vertices in a graph such that no edge connects any two vertices. In adiabatic quantum computation [E. Farhi, et al., Science 292, 472-475 (2001); A. Das, B. K. Chakrabarti, Rev. Mod. Phys. 80, 1061-1081 (2008)], a given graph G(V, E) can be naturally mapped onto a many-body Hamiltonian [Formula: see text], with edges [Formula: see text] being the two-body interactions between adjacent vertices [Formula: see text]. Thus, solving the IS problem is equivalent to finding all the computational basis ground states of [Formula: see text]. Very recently, non-Abelian adiabatic mixing (NAAM) has been proposed to address this task, exploiting an emergent non-Abelian gauge symmetry of [Formula: see text] [B. Wu, H. Yu, F. Wilczek, Phys. Rev. A 101, 012318 (2020)]. Here, we solve a representative IS problem [Formula: see text] by simulating the NAAM digitally using a linear optical quantum network, consisting of three C-Phase gates, four deterministic two-qubit gate arrays (DGA), and ten single rotation gates. The maximum IS has been successfully identified with sufficient Trotterization steps and a carefully chosen evolution path. Remarkably, we find IS with a total probability of 0.875(16), among which the nontrivial ones have a considerable weight of about 31.4%. Our experiment demonstrates the potential advantage of NAAM for solving IS-equivalent problems.

Notes on upper bounds for the largest eigenvalue based on edge-decompositions of a signed graph

The adjacency matrix of a signed graph has +1 or −1 for adjacent vertices, depending on the sign of the connecting edge. According to this concept, an ordinary graph can be interpreted as a signed graph without negative edges. An edge-decomposition of a signed graph Ġ is a partition of its edge set into (non-empty) subsets E1, E2, …, Ek. Every subset Ei (1 ​≤ ​i ​≤ ​k) induces a subgraph of Ġ obtained by keeping only the edges of Ei. Accordingly, a fixed edge-decomposition induces a decomposition of Ġ into the corresponding subgraphs. This paper establishes some upper bounds for the largest eigenvalue of the adjacency matrix of a signed graph Ġ expressed in terms of the largest eigenvalues of subgraphs induced by edge-decompositions. A particular attention is devoted to the so-called cycle decompositions, i.e., decompositions into signed cycles. It is proved that Ġ has a cycle decomposition if and only if it is Eulerian. The upper bounds for the largest eigenvalue in terms of the largest eigenvalues of the corresponding cycles are obtained for regular signed graphs and Hamiltonian signed graphs. These bounds are interesting since all of them can easily be computed, as the largest eigenvalue of a signed cycle is equal to 2 if the product of its edge signs is positive, while otherwise it is 2cosπj, where j stands for the length. Several examples are provided. The entire paper is related to some classical results in which the same approach is applied to ordinary graphs.

Can Romeo and Juliet meet? Or rendezvous games with adversaries on graphs

We introduce the rendezvous game with adversaries. In this game, two players, Facilitator and Divider, play against each other on a graph. Facilitator has two agents and Divider has a team of k agents located in some vertices. They take turns in moving their agents to adjacent vertices (or staying put). Facilitator wins if his agents meet in some vertex. Divider aims to prevent the rendezvous of Facilitator's agents. We show that deciding whether Facilitator can win is PSPACE-hard and, when parameterized by k, co-W[2]-hard. Moreover, even deciding whether Facilitator can win within τ steps is co-NP-complete already for τ=2. On the other hand, for chordal and P5-free graphs, we prove that the problem is solvable in polynomial time. Finally, we show that the problem is fixed-parameter tractable parameterized by both the graph's neighborhood diversity and the number of steps τ.

Sharp Bounds on the Generalized Multiplicative First Zagreb Index of Graphs with Application to QSPR Modeling

Degree sequence measurements on graphs have attracted a lot of research interest in recent decades. Multiplying the degrees of adjacent vertices in graph Ω provides the multiplicative first Zagreb index of a graph. In the context of graph theory, the generalized multiplicative first Zagreb index of a graph Ω is defined as the product of the sum of the αth powers of the vertex degrees of Ω, where α is a real number such that α≠0 and α≠1. The focus of this work is on the extremal graphs for several classes of graphs including trees, unicyclic, and bicyclic graphs, with respect to the generalized multiplicative first Zagreb index. In the initial step, we identify a set of operations that either increases or decreases the generalized multiplicative first Zagreb index for graphs. We then involve analysis of the generalized multiplicative first Zagreb index achieving sharp bounds by characterizing the maximum or minimum graphs for those classes. We present applications of the generalized multiplicative first Zagreb index Π1α for predicting the π-electronic energy Eπ(β) of benzenoid hydrocarbons. In particular, we answer the question concerning the value of α for which the predictive potential of Π1α with Eπ for lower benzenoid hydrocarbons is the strongest. In fact, our statistical analysis delivers that Π1α correlates with Eπ of lower benzenoid hydrocarbons with correlation coefficient ρ=−0.998, if α=−0.00496. In QSPR modeling, the value ρ=−0.998 is considered to be considerably significant.

Connectedness of friends-and-strangers graphs of complete bipartite graphs and others

Let X and Y be any two graphs of order n. The friends-and-strangers graph FS(X,Y) of X and Y is a graph with vertex set consisting of all bijections σ:V(X)↦V(Y), in which two bijections σ, σ′ are adjacent if and only if they differ precisely on two adjacent vertices of X, and the corresponding images are adjacent in Y. The most fundamental question that one can ask about these friends-and-strangers graphs is whether or not they are connected. Let Kk,n−k be a complete bipartite graph of order n. In 1974, Wilson characterized the connectedness of FS(K1,n−1,Y) by using algebraic methods. In this paper, by using combinatorial methods, we investigate the connectedness of FS(Kk,n−k,Y) for any Y and all k≥2, including Y being a random graph, as suggested by Defant and Kravitz, and pose some open problems.

Adjacent Vertex Distinguishing Coloring of Fuzzy Graphs

In this paper, we consider the adjacent vertex distinguishing proper edge coloring (for short, AVDPEC) and the adjacent vertex distinguishing total coloring (for short, AVDTC) of a fuzzy graph. Firstly, this paper describes the development process, the application areas, and the existing review research of fuzzy graphs and adjacent vertex distinguishing coloring of crisp graphs. Secondly, we briefly introduce the coloring theory of crisp graphs and the related theoretical basis of fuzzy graphs, and add some new classes of fuzzy graphs. Then, based on the α-cuts of fuzzy graphs and distance functions, we give two definitions of the AVDPEC of fuzzy graphs, respectively. A lower bound on the chromatic number of the AVDPEC of a fuzzy graph is obtained. With examples, we show that some results of the AVDPEC of a crisp graph do not carry over to our set up; the adjacent vertex distinguishing chromatic number of the fuzzy graph is different from the general chromatic number of a fuzzy graph. We also give a simple algorithm to construct a (d,f)-extended AVDPEC for fuzzy graphs. After that, in a similar way, two definitions of the AVDTC of fuzzy graphs are discussed. Finally, the future research directions of distinguishing coloring of fuzzy graphs are given.

Sigma chromatic number of the middle graph of some general families of trees of height 2

Let [Formula: see text] be a nontrivial connected graph and let [Formula: see text] be a vertex coloring of [Formula: see text] where adjacent vertices may have the same color. For a vertex [Formula: see text] of [Formula: see text] the color sum [Formula: see text] of [Formula: see text] is the sum of the colors of the vertices adjacent to [Formula: see text] The coloring [Formula: see text] is said to be a sigma coloring of [Formula: see text] if [Formula: see text] whenever [Formula: see text] and [Formula: see text] are adjacent vertices in [Formula: see text] The minimum number of colors that can be used in a sigma coloring of [Formula: see text] is called the sigma chromatic number of [Formula: see text] and is denoted by [Formula: see text] In this study, we show that the sigma chromatic number of the middle graph of full binary trees of height [Formula: see text] is 2. We also determine a lower bound for the sigma chromatic number of graphs containing an induced subgraph isomorphic to a complete graph joined by pendant vertices. With this lower bound, we obtain the sigma chromatic number of the middle graph of the graphs [Formula: see text] [Formula: see text] stars, bistars, and the middle graph of some families of trees of height 2.

Application of oriented methods for computing shortest paths and principal curvatures on triangle surfaces

The paper considers two problems: computation of shortest paths on triangle surfaces and calculation of principal curvatures at vertices of triangle surfaces. A technique that allows us to compute smoother and shorter paths using intermediate admissible directions and linear interpolation of potential functions is proposed. Conventional methods that compute the curvature in nodes of triangle surfaces are based on processing only adjacent vertices. They are non stable because triangle meshes contain frequently oscillating errors in positions of the vertices. We propose a simple practical method that accounts for larger neighborhoods of vertices. The kriging method aimed to the sparse representation of specific surface parts (aesthetic regions) is sketched. Preliminary numerical results are presented.

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.