Odd Edge Research Articles

Design and control of topological Fano resonance in Kane-Mele nanoribbons for sensing applications

The concept of topological Fano resonance, characterized by an ultrasharp asymmetric line shape, is a promising candidate for robust sensing applications due to its sensitivity to external parameters and immunity to structural disorder. In this study, the vacancy-induced topological Fano resonance in a nanoribbon made up of a hexagonal lattice with armchair sides is examined by introducing several on-site vacancies, which are deliberately created at regular distances, along a zigzag chain that stretches across the width of the ribbon. The presence of the on-site vacancies can create localized energy states within the electronic band structure, leading to the formation of an impurity band, which can result in Fano resonance phenomena by forming a conductivity channel between the edge modes on both armchair sides of the ribbon. Consequently, an ultracompact tunable on-chip integrated topological Fano resonance derived from the graphene-based nanomechanical phononic crystals is proposed. The Fano resonance arises from the interference between topologically protected even and odd edge modes at the interface between trivial and nontrivial insulators in a nanoribbon structure governed by the Kane-Mele model describing the quantum spin Hall effect in hexagonal lattices. The simulation of the topological Fano resonance is performed analytically using the Lippmann-Schwinger scattering formulation. One of the advantages of the present study is that the related calculations are carried out analytically, and in addition to the simplicity and directness, it reproduces the results obtained from the Landauer-Büttiker formulation very well both quantitatively and qualitatively. The findings open up possibilities for the design of highly sensitive and accurate robust sensors for detecting extremely tiny forces, masses, and spatial positions.

Odd Fibonacci edge irregular labeling for some trees obtained from subdivision and vertex identification operations

Let G be a graph with p vertices and q edges and be an injective function, where k is a positive integer. If the induced edge labeling defined by for each is a bijection, then the labeling f is called an odd Fibonacci edge irregular labeling of G. A graph which admits an odd Fibonacci edge irregular labeling is called an odd Fibonacci edge irregular graph. The odd Fibonacci edge irregularity strength ofes(G) is the minimum k for which G admits an odd Fibonacci edge irregular labeling. In this paper, the odd Fibonacci edge irregularity strength for some subdivision graphs and graphs obtained from vertex identification is determined.

Labeling of 2-regular Graphs by Odd Edge Magic

An edge magic total labeling of a graph G(p, q) is said to be an Odd edge magic total labeling if λ(E) = {1,3, …2q – 1} with the condition that for each edge uv ∈ E, λ(u) + λ(uv) +λ(v) = ke, where ke is known as the magic constant. In this paper we resolute cycles of odd length, disjoint union of C3 ∪ C4r+2 (r ≥ 1), disjoint union of C4 ∪ C4r-1 (r > 1), disjoint union of C3 ∪ C4r (r > 1), disjoint union of C4 ∪ C4r-3 (r > 1), are Odd edge magic graphs.

Modular Version of Edge Irregularity Strength for Fan and Wheel Graphs

A k-labeling from the vertex set of a simple graph G=(V,E) to a set of integers {1,2,…,k} is defined to be a modular edge irregular if, for every couple of distinct edges, their modular edge weights are distinct. The modular edge weight is the remainder of the division of the sum of end vertex labels by modulo |E(G)|. The modular edge irregularity strength of a graph is known as the maximal vertex label k, minimized over all modular edge irregular k-labelings of the graph. In this paper we describe labeling schemes with symmetrical distribution of even and odd edge weights and investigate the existence of (modular) edge irregular labelings of joins of paths and cycles with isolated vertices. We estimate the bounds of the (modular) edge irregularity strength for the join graphs Pn+Km¯ and Cn+Km¯ and determine the corresponding exact value of the (modular) edge irregularity strength for some fan graphs and wheel graphs in order to prove the sharpness of the presented bounds.

Tunable Topological Fano Resonances in Graphene-Based Nanomechanical Lattices

A topological Fano resonance with ultrasharp asymmetric line shape is sensitive to external parameters and is immune to structural disorder, and thus, acts as a robust sensor. Here, an ultracompact tunable on-chip integrated topological Fano resonance is designed based on graphene-based nanomechanical phononic crystals. The Fano resonance comes from the interference between topologically protected even and odd edge modes at the interface between trivial and nontrivial insulators. The two edge modes have different edge-mode widths, resulting in distinct resonant lifetimes. The resonant frequency can be adjusted by employing different gate voltages, which can vary the tension of the graphene membrane. Such tunable topological Fano resonances can be applied to design a wide variety of highly sensitive and accurate robust sensors, including extremely tiny forces, masses, and spatial positions.

Graph Modification for Edge-Coloured and Signed Graph Homomorphism Problems: Parameterized and Classical Complexity

We study the complexity of graph modification problems with respect to homomorphism-based colouring properties of edge-coloured graphs. A homomorphism from edge-coloured graph $G$ to edge-coloured graph $H$ is a vertex-mapping from $G$ to $H$ that preserves adjacencies and edge-colours. We consider the property of having a homomorphism to a fixed edge-coloured graph $H$. The question we are interested in is: given an edge-coloured graph $G$, can we perform $k$ graph operations so that the resulting graph admits a homomorphism to $H$? The operations we consider are vertex-deletion, edge-deletion and switching (an operation that permutes the colours of the edges incident to a given vertex). Switching plays an important role in the theory of signed graphs, that are 2-edge-coloured graphs whose colours are the signs $+$ and $-$. We denote the corresponding problems (parameterized by $k$) by VD-$H$-COLOURING, ED-$H$-COLOURING and SW-$H$-COLOURING. These problems generalise $H$-COLOURING (to decide if an input graph admits a homomorphism to a fixed target $H$). Our main focus is when $H$ is an edge-coloured graph with at most two vertices, a case that is already interesting as it includes problems such as VERTEX COVER, ODD CYCLE RANSVERSAL and EDGE BIPARTIZATION. For such a graph $H$, we give a P/NP-c complexity dichotomy for VD-$H$-COLOURING, ED-$H$-COLOURING and SW-$H$-COLOURING. We then address their parameterized complexity. We show that VD-$H$-COLOURING and ED-$H$-COLOURING for all such $H$ are FPT. In contrast, already for some $H$ of order 3, unless P=NP, none of the three problems is in XP, since 3-COLOURING is NP-c. We show that SW-$H$-COLOURING is different: there are three 2-edge-coloured graphs $H$ of order 2 for which SW-$H$-COLOURING is W-hard, and assuming the ETH, admits no algorithm in time $f(k)n^{o(k)}$. For the other cases, SW-$H$-COLOURING is FPT.

Mechanically controlled quantum switch defined on a curved 2DEG

To investigate quantum nature of two dimensional electrons subject to high perpendicular magnetic fields, usually a planar electronic Fabry-P\'erot interferometer is utilized. In this work, we investigate an interferometer defined on a curved heterostructure. In the presence of a magnetic field perpendicular to the cylindrical axis, the location and the properties of the edge channels depend on the radial component of the magnetic field. Considering a curved structure, we perform numerical and semi-analytical calculations to determine widths of the incompressible edge states. We observe that the edge states form a closed loop for certain magnetic field strengths yielding observation of conductance oscillations, which can be manipulated by changing the Azimuthal angle mechanically. In addition, we investigate the effect of spin polarization on the edge state distribution considering Zeeman splitting and obtained odd integer edge states. The proposed experiment would yield a novel method to clarify the ongoing debate on the origin of conductance oscillations, namely whether they stem from Aharonov-Bohm phase or charging effects

List-coloring – Parameterizing from triviality

The classical graph coloring problem is given an undirected graph and the goal is to color the vertices of the graph with the minimum number of colors so that endpoints of each edge gets different colors. In list-coloring, each vertex is given a list of allowed colors with which it can be colored. Most versions of the problems are hard in several paradigms including approximation and parameterized complexity. We consider a few versions of the problems that are polynomial time solvable and try to extend the notion of feasible algorithms by parameterizing suitably in the paradigm of parameterized complexity. In particular, we provide extensions of these polynomial time solvable coloring problems that belong to the extreme ends of the parameterized complexity hierarchy including fixed-parameter tractable (FPT) class or the complexity class XP or W[1]-hard or para-NP -hard (i.e. NP -hard even for constant values of the parameter).Specifically, we consider generalizations of odd cycle transversal and edge bipartization in the context of list-coloring and show them fixed-parameter tractable. We also look at list coloring where each list has n−k colors and the goal is to color the vertices respecting the list. We show that the problem is in the parameterized complexity class XP, when parameterized by k.

Odd Multiway Cut in Directed Acyclic Graphs

We investigate the odd multiway node (edge) cut problem where the input is a graph with a specified collection of terminal nodes, and the goal is to find a smallest subset of non-terminal nodes (edges) to delete so that the terminal nodes do not have an odd length path between them. In an earlier work, Lokshtanov and Ramanujan showed that both odd multiway node cut and odd multiway edge cut are fixed-parameter tractable (FPT) when parameterized by the size of the solution in undirected graphs. In this work, we focus on directed acyclic graphs (DAGs) and design a fixed-parameter algorithm. Our main contribution is a broadening of the shadow-removal framework to address parity problems in DAGs. We complement our FPT results with tight approximability as well as polyhedral results for two terminals in DAGs. Additionally, we show inapproximability results for odd multiway edge cut in undirected graphs even for two terminals.

((Contractible Edge of Eulerian Graph- Regular ))

In this paper define the contractible edge eulerian graph that, let is a class of Eulerian graphs , the edge e in is called contractible edge eulerian graph if . The necessary conditions for Eulerian graphs to have contractible edge eulerian have been introduced, further, the even and odd contractible edge eulerian graph have been studied , we also define the contractible edge eulerian graph class, the edge e in G is satisfied property contraction is called contractible edge eulerian if . Tutte [7] proved every 3-connected graph non isomorphic to have 3-contractible and proved every 3-connected graph on more than four vertices contains an edge whose contraction yield a new 3-connected graph [7]. We proved graph G is eulerian graph has contractible edge if non isomorphic to . How over every 4-connected graph on at least seven vertices can be reduced to smaller 4-connected graph by contraction one or two edge subsequently [7]. Also we discussed the graph G is eulerian on at least seven vertices can be contraction and saved the properties of eulerian graph. Let be a regular graph and eulerian graph, the edges e in is called contractible regular-eulerian graph if is regular-eulerian grah, We discussed relation contraction of eulerian-regular graph then has contractible if if then has not contractible regular-eulerian.

\(k\)-Odd Edge Mean Labeling of Some Basic Graphs

A \(\left(p,q\right)\) graph \(G\) is said to have a \(k\)-odd edge mean labeling \((k\ge 1)\), if there exists an injection \(f\) from the edges of \(G\) to \(\{0,1,2,3,\ldots,2k+2p-3\}\) such that the induced map \(f^*\) defined on \(V\) by \(f^{*} (v)=\left\lceil \frac{\sum f(vu) }{\deg (v)} \right\rceil\) is a bijection from \(V\) to \(\{2k-1,2k+1,2k+3,\ldots, 2k+2p-3\}\). A graph that admits a \(k\)-odd edge mean labeling is called a \(k\)-odd edge mean graph. In this paper, we have introduced \(k\)-odd edge mean labeling and we have investigated the same labeling for basic graphs like path and star. Also we have examined the existence and non existence of cycles.

A new proof of the flat wall theorem

We give an elementary and self-contained proof, and a numerical improvement, of a weaker form of the excluded clique minor theorem of Robertson and Seymour, the following. Let t,r≥1 be integers, and let R=49152t24(40t2+r). An r-wall is obtained from a 2r×r-grid by deleting every odd vertical edge in every odd row and every even vertical edge in every even row, then deleting the two resulting vertices of degree one, and finally subdividing edges arbitrarily. The vertices of degree two that existed before the subdivision are called the pegs of the r-wall. Let G be a graph with no Kt minor, and let W be an R-wall in G. We prove that there exist a set A⊆V(G) of size at most 12288t24 and an r-subwall W′ of W such that V(W′)∩A=∅ and W′ is a flat wall in G−A in the following sense. There exists a separation (X,Y) of G−A such that X∩Y is a subset of the vertex set of the cycle C′ that bounds the outer face of W′, V(W′)⊆Y, every peg of W′ belongs to X and the graph G[Y] can almost be drawn in the unit disk with the vertices X∩Y drawn on the boundary of the disk in the order determined by C′. Here almost means that the assertion holds after repeatedly removing parts of the graph separated from X∩Y by a cutset Z of size at most three, and adding all edges with both ends in Z. Our proof gives rise to an algorithm that runs in polynomial time even when r and t are part of the input instance. The proof is self-contained in the sense that it uses only results whose proofs can be found in textbooks.

On the inertia set of a signed tree with loops

A signed graph is a pair (G,Σ), where G=(V,E) is a graph (in which parallel edges and loops are permitted) with V={1,…,n} and Σ⊆E. The edges in Σ are called odd edges and the other edges of E even. By S(G,Σ) we denote the set of all n×n real symmetric matrices A=[ai,j] such that if ai,j<0, then among the edges connecting i and j, there must be at least one even edge; if ai,j>0, then among the edges connecting i and j, there must be at least one odd edge; and if ai,j=0, then either there must be at least one odd edge and at least one even edge connecting i and j, or there are no edges connecting i and j. (Here we allow i=j.) For a real symmetric matrix A, the partial inertia of A is the pair (p,q), where p and q are the number of positive and negative eigenvalues of A, respectively. If (G,Σ) is a signed graph, we define the inertia set of (G,Σ) as the set of the partial inertias of all matrices A∈S(G,Σ). By MR(G,Σ) we denote max⁡{rank(A)|A∈S(G,Σ)}. We say that a signed graph (G,Σ) satisfies the Northeast Property if for each (p,q) with p+q<MR(G,Σ) in the inertia set of (G,Σ), also (p+1,q),(p,q+1) belong to the inertia set of (G,Σ).In this paper, we show that if (G,Σ) is a signed graph, where G is a tree with possibly loops attached at some of the vertices, then (G,Σ) satisfies the Northeast Property. Furthermore, we present a formula for calculating the inertia set of a signed graph (G,Σ), where G is a tree with possibly loops attached at some of the vertices.

On the Minimum Number of Spanning Trees ink-Edge-Connected Graphs

We show that a k-edge-connected graph on n vertices has at least n(k/2)n−1 spanning trees. This bound is tight if k is even and the extremal graph is the n-cycle with edge multiplicities k/2. For k odd, however, there is a lower bound ckn−1, where ck>k/2. Specifically, c3>1.77 and c5>2.75. Not surprisingly, c3 is smaller than the corresponding number for 4-edge-connected graphs. Examples show that c3≤2+3≈1.93. However, we have no examples of 5-edge-connected graphs with fewer spanning trees than the n-cycle with all edge multiplicities (except one) equal to 3, which is almost 6-regular. We have no examples of 5-regular 5-edge-connected graphs with fewer than 3.09n−1 spanning trees, which is more than the corresponding number for 6-regular 6-edge-connected graphs. The analogous surprising phenomenon occurs for each higher odd edge connectivity and regularity.

Hecke algebras with independent parameters

We study the Hecke algebra $\H(\bq)$ over an arbitrary field $\FF$ of a Coxeter system $(W,S)$ with independent parameters $\bq=(q_s\in\FF:s\in S)$ for all generators. This algebra is always linearly spanned by elements indexed by the Coxeter group $W$. This spanning set is indeed a basis if and only if every pair of generators joined by an odd edge in the Coxeter diagram receive the same parameter. In general, the dimension of $\H(\bq)$ could be as small as $1$. We construct a basis for $\H(\bq)$ when $(W,S)$ is simply laced. We also characterize when $\H(\bq)$ is commutative, which happens only if the Coxeter diagram of $(W,S)$ is simply laced and bipartite. In particular, for type A we obtain a tower of semisimple commutative algebras whose dimensions are the Fibonacci numbers. We show that the representation theory of these algebras has some features in analogy/connection with the representation theory of the symmetric groups and the 0-Hecke algebras.

Signed graphs whose signed Colin de Verdière parameter is two

A signed graph is a pair (G,Σ), where G=(V,E) is a graph (in which parallel edges are permitted, but loops are not) with V={1,…,n} and Σ⊆E. The edges in Σ are called odd and the other edges even. By S(G,Σ) we denote the set of all real symmetric n×n matrices A=[ai,j] with ai,j<0 if i and j are adjacent and all edges between i and j are even, ai,j>0 if i and j are adjacent and all edges between i and j are odd, and ai,j=0 if i≠j and i and j are non-adjacent. The parameter ν(G,Σ) of a signed graph (G,Σ) is the largest nullity of any positive semidefinite matrix A∈S(G,Σ) that has the Strong Arnold Property. By K3= we denote the signed graph obtained from (K3,∅) by adding to each even edge an odd edge in parallel. In this paper, we prove that a signed graph (G,Σ) has ν(G,Σ)≤2 if and only if (G,Σ) has no minor isomorphic to (K4,E(K4)) or K3=.

On the inertia set of a signed graph with loops

A signed graph is a pair (G,Σ), where G=(V,E) is a graph (in which parallel edges and loops are permitted) with V={1,…,n} and Σ⊆E. The edges in Σ are called odd edges and the other edges of E even. By S(G,Σ) we denote the set of all symmetric n×n real matrices A=[ai,j] such that if ai,j<0, then among the edges connecting i and j, there must be at least one even edge; if ai,j>0, then among the edges connecting i and j, there must be at least one odd edge; and if ai,j=0, then either there must be at least one odd edge and at least one even edge connecting i and j, or there are no edges connecting i and j. (Here we allow i=j.) For a symmetric real matrix A, the partial inertia of A is the pair (p,q), where p and q are the numbers of positive and negative eigenvalues of A, respectively. If (G,Σ) is a signed graph, we define the inertia set of (G,Σ) as the set of the partial inertias of all matrices A∈S(G,Σ).In this paper, we present formulas that allows us to obtain the minimal elements of the inertia set of (G,Σ) in case (G,Σ) has a 1-separation using the inertia sets of certain signed graphs associated with the 1-separation.

Odd edge coloring of graphs

An edge coloring of a graph G is said to be an odd edge coloring if for each vertex v of G and each color c, the vertex v uses the color c an odd number of times or does not use it at all. In [5], Pyber proved that 4 colors suffice for an odd edge coloring of any simple graph. Recently, some results on this type of colorings of (multi)graphs were successfully applied in solving a problem of facial parity edge coloring [3, 2]. In this paper we present additional results, namely we prove that 6 colors suffice for an odd edge coloring of any loopless connected (multi)graph, provide examples showing that this upper bound is sharp and characterize the family of loopless connected (multi)graphs for which the bound 6 is achieved. We also pose several open problems.

Nowhere-zero 3-flows of graphs with prescribed sizes of odd edge cuts

The relation between sizes of odd-cycles and chromatic numbers of graphs has been extensively studied. It was conjectured by Bollobás and Erdős (1990) [4] and proved by Gyárfás [A. Gyárfás, Graph with k odd cycle lengths, Discrete Math. 103 (1992) 41–48] that if a graphGhaskdifferent odd cycle sizes, then the chromatic number ofG, χ(G)≤2k+2. Recently, Camacho and Schiermeyer [S.M. Camacho, I. Schiermeyer, Colorings of graphs with two consecutive odd cycle lengths, Discrete Math. 309 (2009) 4916–4919] proved that every graph containing only odd cycles of lengths2k−1and2k+1is 4-colorable. Wang [S.S. Wang, Structure and coloring of graphs with only small odd cycles, SIAM J. Discrete Math. 22 (2008) 1040–1072] characterizes all 3-colorable graphs whose only odd cycles are 3-cycles and 5-cycles; and Kaiser et al. [T. Kaiser, O. Rucký, R. Škrekovski, Graphs with odd cycle lengths 5 and 7 are 3-colorable, SIAM J. Discrete Math. 25 (2012) 1069–1088] proved that every graph with odd cycle lengths 5 and 7 is 3-colorable.In this paper, we investigate the dual version of this problem: the relation between integer flows and odd edge cuts (the integer flow problem is a dual version of the vertex coloring problem, and the dual of an odd-cycle is an odd-edge-cut). The main result in this paper shows that, for each odd integer k≥3, if the size of every odd edge cut of a graph G is in {k,k+2,…,3k−2}, then G admits a nowhere-zero 3-flow if and only if G cannot be reduced to three well-defined families of graphs.

Insights into the mechanism of interaction between trehalose-conjugated beta-sheet breaker peptides and Aβ(1–42) fibrils by molecular dynamics simulations

An attractive strategy to contrast the Alzheimer disease (AD) is represented by the development of β-sheet breaker peptides (BSB). β-sheet breakers constitute a class of compounds which have shown a good efficacy in preventing the Aβ fibrillogenesis; however, their mechanism of action has not been precisely understood. In this context, we have studied the structural basis underlying the inhibitory effect of Aβ(1-42) fibrillogenesis explicated by two promising trehalose-conjugated BSB peptides using an all-atom molecular dynamics (MD) approach. Our simulations suggest that the binding on the two protofibril ends occurs through different binding modes. In particular, binding on the odd edge (chain A) is guided by a well defined hydrophobic cleft, which is common to both ligands. Moreover, targeting chain A entails a significant structure destabilization leading to a partial loss of β structure and is an energetically favoured process. A significant contribution of the trehalose moiety to the stability of the complexes emerged from our results. The energetically favoured hydrophobic cleft detected on chain A could represent a good starting point for the design of new molecules with improved anti-aggregating features.