Edge Joining Research Articles

Polynomial lower bound on the effective resistance for the one‐dimensional critical long‐range percolation

AbstractIn this work, we study the critical long‐range percolation (LRP) on , where an edge connects and independently with probability 1 for and with probability for some fixed . Viewing this as a random electric network where each edge has a unit conductance, we show that with high probability the effective resistances from the origin 0 to and from the interval to (conditioned on no edge joining and ) both have a polynomial lower bound in . Our bound holds for all and thus rules out a potential phase transition (around ) which seemed to be a reasonable possibility.

Multilevel Feature Fusion for End-to-End Blind Image Quality Assessment

In this paper, a framework based on two feature extraction networks and a multilevel feature fusion (MFF) network is proposed. Multilevel degradation features can be obtained through this method, and combined with the human visual perception system, the local and global feature information contained in these features can be captured, which is conducive to the prediction of distorted images. First, a restored image approximating a reference image is generated by a restorative generative adversarial network (GAN). Furthermore, the multilevel degradation features of distorted images and the restored image features are extracted by EfficientNet. Second, the features extracted by EfficientNet are input into the MFF network and are fully expressed by the top-down, bottom-up and third edge joining methods. Moreover, the features provide more low-level details and high-level semantic features for the prediction of image quality scores. In addition, after the MFF stage, the framework calculates the score of each branch feature and obtains the average quality score. Experimental results show that our method achieves greatly improved prediction accuracy and performance on five standard databases.

On Reconfiguration Graphs of Independent Sets Under Token Sliding

An independent set of a graph G is a vertex subset I such that there is no edge joining any two vertices in I. Imagine that a token is placed on each vertex of an independent set of G. The $${\textsf{TS}}$$ - ( $${\textsf{TS}}_k$$ -) reconfiguration graph of G takes all non-empty independent sets (of size k) as its nodes, where k is some given positive integer. Two nodes are adjacent if one can be obtained from the other by sliding a token on some vertex to one of its unoccupied neighbors. This paper focuses on the structure and realizability of these reconfiguration graphs. More precisely, we study two main questions for a given graph G: (1) Whether the $${\textsf{TS}}_k$$ -reconfiguration graph of G belongs to some graph class $${\mathcal {G}}$$ (including complete graphs, paths, cycles, complete bipartite graphs, connected split graphs, maximal outerplanar graphs, and complete graphs minus one edge) and (2) If G satisfies some property $${\mathcal {P}}$$ (including s-partitedness, planarity, Eulerianity, girth, and the clique’s size), whether the corresponding $${\textsf{TS}}$$ - ( $${\textsf{TS}}_k$$ -) reconfiguration graph of G also satisfies $${\mathcal {P}}$$ , and vice versa. Additionally, we give a decomposition result for splitting a $${\textsf{TS}}_k$$ -reconfiguration graph into smaller pieces.

Spectral radius conditions for the existence of all subtrees of diameter at most four

Let μ(G) denote the spectral radius of a graph G. We partly confirm a conjecture due to Nikiforov, which is a spectral radius analogue of the well-known Erdős-Sós Conjecture that any tree of order t is contained in a graph of average degree greater than t−2. Let Sn,k be the graph obtained by joining every vertex of a complete graph on k vertices to every vertex of an independent set of order n−k, and let Sn,k+ be the graph obtained from Sn,k by adding a single edge joining two vertices of the independent set of Sn,k. In 2010, Nikiforov conjectured that for a given integer k, every graph G of sufficiently large order n with μ(G)≥μ(Sn,k+) contains all trees of order 2k+3, unless G=Sn,k+. We confirm this conjecture for trees with diameter at most four, with one exception. In fact, we prove the following stronger result for k≥8. If a graph G with sufficiently large order n satisfies μ(G)≥μ(Sn,k) and G≠Sn,k, then G contains all trees of order 2k+3 with diameter at most four, except for the tree obtained from a star on k+2 vertices by subdividing each of its k+1 edges once.

Scaling limit of linearly edge-reinforced random walks on critical Galton-Watson trees

We prove an invariance principle for linearly edge reinforced random walks on γ-stable critical Galton-Watson trees, where γ∈(1,2] and where the edge joining x to its parent has rescaled initial weight d(O,x)α for some α≤1. This corresponds to the recurrent regime of initial weights. We then establish fine asymptotics for the limit process. In the transient regime, we also give an upper bound on the random walk displacement in the discrete setting, showing that the edge reinforced random walk never has positive speed, even when the initial edge weights are strongly biased away from the root.

Gallai–Ramsey number of odd cycles with chords

Gallai–Ramsey number of odd cycles with chords

Gallai-Ramsey number of even cycles with chords

Gallai-Ramsey number of even cycles with chords

On automorphism group of orthogonality graph of finite semisimple rings

The orthogonality graph of a ring R, written as O(R), is an undirected simple graph with vertex set the set of all nonzero two-sides zero-divisors of R, and for distinct there exists an edge joining x and y if and only if In this paper, we describe the automorphism group of O(R) when R is a finite semisimple ring.

ON THE AUTOMORPHISMS GROUP OF FINITE POWER GRAPHS

The power graph of a group $G$ is the graph with vertex set $G$,having an edge joining $x$ and $y$ whenever one is a power of theother. The purpose of this paper is to study the automorphismgroups of the power graphs of infinite groups.

C4C8(S) tori which are Cayley graphs

A [Formula: see text] net is a trivalent decoration made by alternating square [Formula: see text] and octagons [Formula: see text]. It can cover either a cylinder or a tori. Cayley graph [Formula: see text] on a group [Formula: see text] with connection set [Formula: see text] has the elements of [Formula: see text] as its vertices and an edge joining [Formula: see text] and [Formula: see text] for all [Formula: see text] and [Formula: see text]. Motivated by Afshari’s work, we show that the [Formula: see text] tori are Cayley graphs by constructing a regular subgroup of the automorphism group of [Formula: see text].

Competition Numbers and Phylogeny Numbers of Connected Graphs and Hypergraphs

Let D be a digraph. The competition graph of D is the graph having the same vertex set with D and having an edge joining two different vertices if and only if they have at least one common out-neighbor in D. The phylogeny graph of D is the competition graph of the digraph constructed from D by adding loops at all vertices. The competition/phylogeny number of a graph is the least number of vertices to be added to make the graph a competition/phylogeny graph of an acyclic digraph. In this paper, we show that for any integer k there is a connected graph such that its phylogeny number minus its competition number is greater than k. We get similar results for hypergraphs.

Upper density of monochromatic infinite paths

Ramsey Theory investigates the existence of large monochromatic substructures. Unlike the most classical case of monochromatic complete subgraphs, the maximum guaranteed length of a monochromatic path in a two-edge-colored complete graph is well-understood. Gerencsér and Gyárfás in 1967 showed that any two-edge-coloring of a complete graph Kn contains a monochromatic path with ⌊2n/3⌋+1 vertices. The following two-edge-coloring shows that this is the best possible: partition the vertices of Kn into two sets A and B such that |A|=⌊n/3⌋ and |B|=⌈2n/3⌉, and color the edges between A and B red and edges inside each of the sets blue. The longest red path has 2|A|+1 vertices and the longest blue path has |B| vertices. The main result of this paper concerns the corresponding problem for countably infinite graphs. To measure the size of a monochromatic subgraph, we associate the vertices with positive integers and consider the lower and the upper density of the vertex set of a monochromatic subgraph. The upper density of a subset A of positive integers is the limit superior of |A∩{1,...,}|/n, and the lower density is the limit inferior. The following example shows that there need not exist a monochromatic path with positive upper density such that its vertices form an increasing sequence: an edge joining vertices i and j is colored red if ⌊log2i⌋≠⌊log2j⌋, and blue otherwise. In particular, the coloring yields blue cliques with 1, 2, 4, 8, etc., vertices mutually joined by red edges. Likewise, there are constructions of two-edge-colorings such that the lower density of every monochromatic path is zero. A result of Rado from the 1970's asserts that the vertices of any k-edge-colored countably infinite complete graph can be covered by k monochromatic paths. For a two-edge-colored complete graph on the positive integers, this implies the existence of a monochromatic path with upper density at least 1/2. In 1993, Erdős and Galvin raised the problem of determining the largest c such that every two-edge-coloring of the complete graph on the positive integers contains a monochromatic path with upper density at least c. The authors solve this 25-year-old problem by showing that c=(12+8–√)/17≈0.87226.

Extremal Theta-free planar graphs

Extremal Theta-free planar graphs

Gutman and Degree Monophonic Index of Graphs

A graph with p points and q edges is denoted by G(p,q). An edge joining two non-adjacent points of a path P is called a chord of a path P. A path P is called monophonic if it is a chordless path. For any two points u and v in a connected graph G, the monophonic distance d (u,v) m from u to v is defined as the length of a longest u-v monophonic path in G. The Gutman monophonic index of a graph G is denoted by GutMP(G) and defined by GutMP(G) d(u)d(v)d (u,v) m and degree monophonic index of G is denoted by DMP(G) and defined by DMP(G) d(u) d(v)d (u,v)   m . The methodology executed in this research paper is to determine the monophonic distance matrix of graphs under consideration. The entries of monophonic distance matrix are calculated by counting the number of edges in the u-v monophonic path. In this paper for some standard graphs, GutMP(G) and DMP(G) are studied which can be applied to derive quantitative structure- property or structure- activity relationships (QSPR / QSAR).

Partite Saturation of Complete Graphs

We study the problem of determining sat$(n,k,r)$, the minimum number of edges in a $k$-partite graph $G$ with $n$ vertices in each part such that $G$ is $K_r$-free but the addition of an edge joining any two nonadjacent vertices from different parts creates a $K_r$. Improving recent results of Ferrara, Jacobson, Pfender, and Wenger, and generalizing a recent result of Roberts, we show that sat$(n,k,r) = \alpha(k,r)n + O(1)$ as $n \rightarrow \infty$. Moreover, for every $3 \leq r \leq k$ we prove that $k(2k - 4) \leq \alpha(k,r) \leq (k-1)(4 r - \ell -6),$ where $\ell = \mbox{min}\{k, 2r-3\}$, and show that the lower bound is tight for infinitely many values of $r$ and every $k\geq 2r-1$. This allows us to prove that, for these values, sat$(n,k,r) = k(2r-4)n + O(1)$ as $n \rightarrow \infty$. Along the way, we disprove a conjecture and answer a question of the first set of authors mentioned above.

Diffusion bonding of a titanium alloy to a stainless steel with an aluminium alloy interlayer

Diffusion bonding is the process of joining similar or dissimilar materials at the interface rather than edge joining using welding process or other metal joining processes. It is a solid state welding process carried out under the conditions of soaking temperature, bonding time and load on the surfaces to be joined. Generally this is carried out under vacuum or inert atmosphere to avoid interface reactions to actuate the corrosion. Diffusion bonding process done at different conditions of temperature, bonding time and load. These parameters are optimised for better bond strength, the bonded specimens are analysed for microstructural changes and bond strength since diffusion bonding is an atomic transfer between the surfaces. The micro hardness test is conducted to evaluate the bond strength at the interface. The strength of the bond can be improved by inserting another material at the interface as an interlayer. The base metal titanium with stainless steel is bonded with aluminium as an interlayer.

The total detour monophonic number of a graph

For a connected graph G = (V, E) of order at least two, a chord of a path P is an edge joining two non-adjacent vertices of P. A path P is called a monophonic path if it is a chordless path. A longest x – y monophonic path is called an x – y detour monophonic path. A set S of vertices of G is a detour monophonic set of G if each vertex v of G lies on an x - y detour monophonic path for some x and y in S. The minimum cardinality of a detour monophonic set of G is the detour monophonic number of G and is denoted by dm(G). A total detour monophonic set of a graph G is a detour monophonic set S such that the subgraph induced by S has no isolated vertices. The minimum cardinality of a total detour monophonic set of G is the total detour monophonic number of G and is denoted by dmt(G). A total detour monophonic set of cardinality dmt(G) is called a dmt-set of G. We determine bounds for it and characterize graphs which realize the lower bound. It is shown that for positive integers r, d and k ≥ 6 with r < d there exists a connected graph G with monophonic radius r, monophonic diameter d and dmt(G) = k. For positive integers a, b such that 4 ≤ a ≤ b with b ≤ 2a, there exists a connected graph G such that dm(G) = a and dmt(G) = b. Also, if p, d and k are positive integers such that 2 ≤ d ≤ p - 2, 3 ≤ k ≤ p and p – d – k + 3 ≥ 0, there exists a connected graph G of order p, monophonic diameter d and dmt(G) = k.

Linear transformation distance for bichromatic matchings

Linear transformation distance for bichromatic matchings

Extreme Monophonic Graphs and Extreme Geodesic Graphs

For a connected graph $G=(V,E)$ of order at least two, a chord of a path $P$ is an edge joining two non-adjacent vertices of $P$. A path $P$ is called a monophonic path if it is a chordless path. A monophonic set of $G$ is a set $S$ of vertices such that every vertex of $G$ lies on a monophonic path joining some pair of vertices in $S$. The monophonic number of $G$ is the minimum cardinality of its monophonic sets and is denoted by $m(G)$. A geodetic set of $G$ is a set $S$ of vertices such that every vertex of $G$ lies on a geodesic joining some pair of vertices in $S$. The geodetic number of $G$ is the minimum cardinality of its geodetic sets and is denoted by $g(G)$. The number of extreme vertices in $G$ is its extreme order $ex(G)$. A graph $G$ is an extreme monophonic graph if $m(G)=ex(G)$ and an extreme geodesic graph if $g(G)=ex(G)$. Extreme monophonic graphs of order $p$ with monophonic number $p$ and $p-1$ are characterized. It is shown that every pair $a,b$ of integers with $0 \leq a \leq b$ is realized as the extreme order and monophonic number, respectively, of some graph. For positive integers $r,d$ and $k \geq 3$ with $r &lt; d$, it is shown that there exists an extreme monophonic graph $G$ of monophonic radius $r$, monophonic diameter $d$, and monophonic number $k$. Also, we give a characterization result for a graph $G$ which is both extreme geodesic and extreme monophonic.

Upper Detour Monophonic Number of a Graph

Upper Detour Monophonic Number of a Graph