Internal Vertices Research Articles

Determination of particular double starlike trees by the Laplacian spectrum

A double starlike tree is a tree in which exactly two vertices have degree greater than two. In this study we consider double starlike trees obtained by attaching p−2(forp≥3) pendant vertices at an internal vertex and q−2(q≥3) pendant vertices at a different internal vertex of a fixed path P. We denote this tree by T≅D(a,b,c,p,q), where a,b,c stand for the numbers of vertices in segments of P obtained by deleting vertices of degree p and q. It is known that, depending on parameters, T may or may not be determined by its Laplacian spectrum. In the latter case we provide the structure of a putative tree with the same Laplacian spectrum. This result implies the known result stating that D(1,b,1,p,q) is determined by the Laplacian spectrum and two new results stating the same for D(1,b,2,p,p) and D(2,b,2,p,p).

Triangle Counting Rule: An Approach to Forecast the Magnetic Properties of Benzenoid Polycyclic Hydrocarbons.

Open-shell benzenoid polycyclic hydrocarbons (BPHs) are promising materials for future quantum applications. However, the search for and realization of open-shell BPHs with desired properties is a challenging task due to the gigantic chemical space of BPHs, requiring new strategies for both theoretical understanding and experimental advancement. In this work, by building a structure database of BPHs through graphical enumeration, performing data-driven analysis, and combining tight-binding and mean-field Hubbard calculations, we discovered that the number of the internal vertices of the BPH graphs is closely correlated to their open-shell characters. We further established a simple rule, the triangle counting rule, to predict the magnetic ground states of BPHs. These findings not only provide a database of open-shell BPHs, but also extend the well-known Lieb's theorem and Ovchinnikov's rule and provide a straightforward method for designing open-shell carbon nanostructures. These insights may aid in the exploration of emerging quantum phases and the development of magnetic carbon materials for technology applications.

Triangular-grid billiards and plabic graphs

Given a polygon \(P\) in the triangular grid, we obtain a permutation \(\pi_P\) via a natural billiards system in which beams of light bounce around inside of \(P\). The different cycles in \(\pi_P\) correspond to the different trajectories of light beams. We prove that \[\operatorname{area}(P)\geq 6\operatorname{cyc}(P)-6\quad\text{and}\quad\operatorname{perim}(P)\geq\frac{7}{2}\operatorname{cyc}(P)-\frac{3}{2},\] where \(\operatorname{area}(P)\) and \(\operatorname{perim}(P)\) are the (appropriately normalized) area and perimeter of \(P\), respectively, and \(\operatorname{cyc}(P)\) is the number of cycles in \(\pi_P\). The inequality concerning \(\operatorname{area}(P)\) is tight, and we characterize the polygons \(P\) satisfying \(\operatorname{area}(P)=6\operatorname{cyc}(P)-6\). These results can be reformulated in the language of Postnikov's plabic graphs as follows. Let \(G\) be a connected reduced plabic graph with essential dimension \(2\). Suppose \(G\) has \(n\) marked boundary points and \(v\) (internal) vertices, and let \(c\) be the number of cycles in the trip permutation of \(G\). Then we have \[v\geq 6c-6\quad\text{and}\quad n\geq\frac{7}{2}c-\frac{3}{2}.\]Mathematics Subject Classifications: 05D99, 51M04Keywords: Triangular grid, billiards, plabic graph, membrane

Controllability for the wave equation on graph with cycle and delta-prime vertex conditions

Exact controllability for the wave equation on a metric graph consisting of a cycle and two attached edges is proven. One boundary and one internal control are used. At the internal vertices, delta-prime conditions are satisfied. As a second example, we examine a tripod controlled at the root and the junction, while the leaves are fixed. These examples are key to understanding controllability properties in general metric graphs.

Low-Complexity and High-Coding-Efficiency Image Deletion for Compressed Image Sets in Cloud Servers

Image deletion refers to removing images from a compressed image set in cloud servers, which has always received much attention. However, in some cases images are not successfully deleted, and coding performance still remains to rise. In this paper, we propose a low-complexity and high-coding-efficiency image deletion algorithm. First, all the images are classified into to-be-deleted images, images unneeded to be processed, and images needed to be processed further divided into images needed to be only decoded and images needed to be re-encoded. Then, we also propose a depth- and subtree-constrained minimum spanning tree (DSCMST) heuristics to produce the DSCMST of images needed to be processed. Third, every image unneeded to be processed is added to the just obtained DSCMST as the child of the vertex that is still its parent in the compressed image set. Finally, after the encoding of images needed to be re-encoded, a new compressed image set is constructed, implying the completion of image deletion. Experimental results show that under various circumstances our proposed algorithm can effectively remove any images, including root vertex, internal vertices, and leaf vertices. Moreover, compared with state-of-the-art methods, the proposed algorithm achieves higher coding efficiency while having the minimum complexity.

Motives of melonic graphs

We investigate recursive relations for the Grothendieck classes of the affine graph hypersurface complements of melonic graphs. We compute these classes explicitly for several families of melonic graphs, focusing on the case of graphs with valence- 4 internal vertices, relevant to CTKT tensor models. The results hint at a complex and interesting structure in terms of divisibility relations or nontrivial relations between classes of graphs in different families. Using the recursive relations, we prove that the Grothendieck classes of all melonic graphs are positive as polynomials in the class of the moduli space \mathcal M_{0,4} . We also conjecture that the corresponding polynomials are log-concave , on the basis of hundreds of explicit computations.

Developing A Secure Cryptosystem with Rainbow Vertex Antimagic Coloring of Cycle Graph

An edge labeling of graph G is a function g from the edge set of graph G to the first natural numbers up to the number of the edge set. Graph G admits a rainbow vertex antimagic coloring if, for any two vertices, there is a path with different colors of all internal vertices. The vertex color of graph G is assigned by vertex weight. The vertex weight of graph G is obtained by summing all edge labels that incident with that vertex. The rainbow vertex antimagic connection number of graph G, denoted by rvac(G) is the smallest number of different colors induced by rainbow vertex antimagic coloring. In this research, we determine the upper bound of the rainbow vertex antimagic connection number (rvac) on a cycle graph (Cn) and create a secured cryptosystem using a modified Affine Cipher based on rainbow vertex antimagic coloring.

Bounds on the connectivity of iterated line graphs

For simple connected graphs that are neither paths nor cycles, we define l ( G )=max{ m : G has a divalent path of length m that is not both of length 2 and in a K 3 } , where a divalent path is a path whose internal vertices have degree two in G . Let G be a graph and L n ( G ) be its n -th iterated line graph of G . We use κ e ′( G ) and κ ( G ) for the essential edge connectivity and vertex connectivity of G , respectively. Let G be a simple connected graph that is not a path, a cycle or K 1, 3 , with l ( G )= l ≥ 1 . We prove that (i) for integers s ≥ 1 , κ ′ e ( L l + s ( G )) ≥ 2 s + 2 ; (ii) for integers s ≥ 2 , κ ( L l + s ( G )) ≥ 2 s − 1 + 2 . The bounds are best possible.

The Threshold for Stacked Triangulations

Abstract A stacked triangulation of a $d$-simplex $\mathbf {o}=\{1,\ldots ,d+1\}$ ($d\geq 2$) is a triangulation obtained by repeatedly subdividing a $d$-simplex into $d+1$ new ones via a new vertex (the case $d=2$ is known as an Appolonian network). We study the occurrence of such a triangulation in the Linial–Meshulam model, that is, for which $p$ does the random simplicial complex $Y\sim {\mathcal {Y}}_d(n,p)$ contain the faces of a stacked triangulation of the $d$-simplex $\mathbf {o}$, with its internal vertices labeled in $[n]$. In the language of bootstrap percolation in hypergraphs, it pertains to the threshold for $K_{d+2}^{d+1}$, the $(d+1)$-uniform clique on $d+2$ vertices. Our main result identifies this threshold for every $d\geq 2$, showing it is asymptotically $(\alpha _d n)^{-1/d}$, where $\alpha _d$ is the growth rate of the Fuss–Catalan numbers of order $d$. The proof hinges on a second moment argument in the supercritical regime and on Kalai’s algebraic shifting in the subcritical regime.

Community Evolution Prediction Based on Multivariate Feature Sets and Potential Structural Features

The current study on community evolution prediction ignores the problem of internal community topology characteristics and does not take feature sets extraction into account. Therefore, the MF-PSF (Multivariate Feature sets and Potential Structural Features) method based on multivariate feature sets and potential structural features for community evolution prediction is proposed in this paper. Firstly, the multivariate feature sets are built from four aspects: community core node features, community structural features, community sequential features and community behavior features. Secondly, the community’s potential structural characteristics based on DeepWalk and spectral propagation theories are extracted, and the overall community’s internal structural characteristics and vertex distribution are analyzed. Finally, the community’s multivariate structural features and potential structural features are merged to predict community evolution events, and the importance of each feature in the process of evolutionary prediction is discussed. The experimental results show that compared with other community evolution prediction methods, the MF-PSF prediction method not only provides a foundation for analyzing the influence of various feature sets on predicted events, but it also effectively improves the accuracy of evolution prediction.

Algorithms for maximum internal spanning tree problem for some graph classes

For a given graph G, a maximum internal spanning tree of G is a spanning tree of G with maximum number of internal vertices. The Maximum Internal Spanning Tree (MIST) problem is to find a maximum internal spanning tree of the given graph. The MIST problem is a generalization of the Hamiltonian path problem. Since the Hamiltonian path problem is NP-hard, even for bipartite and chordal graphs, two important subclasses of graphs, the MIST problem also remains NP-hard for these graph classes. In this paper, we propose linear-time algorithms to compute a maximum internal spanning tree of cographs, block graphs, cactus graphs, chain graphs and bipartite permutation graphs. The optimal path cover problem, which asks to find a path cover of the given graph with maximum number of edges, is also a well studied problem. In this paper, we also study the relationship between the number of internal vertices in maximum internal spanning tree and number of edges in optimal path cover for the special graph classes mentioned above.

The Independence Number Conditions for 2-Factors of a Claw-Free Graph

In 2014, some scholars showed that every 2-connected claw-free graph G with independence number α(G)≤3 is Hamiltonian with one exception of family of graphs. If a nontrivial path contains only internal vertices of degree two and end vertices of degree not two, then we call it a branch. A set S of branches of a graph G is called a branch cut if we delete all edges and internal vertices of branches of S leading to more components than G. We use a branch bond to denote a minimal branch cut. If a branch-bond has an odd number of branches, then it is called odd. In this paper, we shall characterize all 2-connected claw-free graphs G such that every odd branch-bond of G has an edge branch and such that α(G)≤5 but has no 2-factor. We also consider the same problem for those 2-edge-connected claw-free graphs with α(G)≤4.

Intelligent computational design of scalene-faceted flat-foldable tessellations

Abstract Origami tessellations can be folded from a given planar pattern into a three-dimensional object with specific geometric properties, inspiring developments in various fields of science and engineering such as deployable structures, energy absorption devices, reconfigurable robots, and metamaterials. However, the range of existing origami patterns with functional properties such as flat-foldability is rather scant, as analytical solutions to constraint equations arising in the design process are generally highly complicated. In this paper, we tackle the challenging problem of automated design of scalene-faceted flat-foldable origami tessellations using an efficient metaheuristic algorithm. To this end, this study establishes constraint curves based on compatibility conditions for all six-fold (i.e., degree-6) vertices. Subsequently, a graphical method and a particle swarm optimization (PSO) method are adopted to produce optimal origami patterns. Moreover, mountain-valley assignments for the obtained geometric designs are determined using a computational approach based on mixed-integer linear programming. It turns out that the flat-foldable internal vertices of each C2-symmetric unit fragment (UF) exist as C2-symmetric pairs about the centroid of the UF. Furthermore, numerical experiments are carried out to examine the feasibility and compare the accuracy, computational efficiency, and global convergence of the proposed methods. The results of numerical experiments demonstrated that, in comparison with the graphical method, the proposed PSO method has not only a higher accuracy but also a significantly lower computational cost, enabling us to develop an intelligent computational platform to efficiently design scalene-faceted flat-foldable origami tessellations.

On the Planarity of a Directed Pathos Total Digraphs of Some Special Arborescence Graphs

An arborescence graph is a directed graph in which, for a vertex u called the root, and any other vertex v, there is exactly one directed path from u to v. The directed pathos of an arborescence Ar is defined as a collection of minimum number of arc disjoint open directed paths whose union is Ar. In [6], for an arborescence Ar, a directed pathos total digraph Q = DP T(Ar)has vertex set V (Q) = V (Ar) ∪ A(Ar) ∪ P(Ar), where V (Ar) is the vertex set, A(Ar) is the arc set, and P(Ar) is a directed pathos set of Ar. The arc set A(Q) consists of the following arcs: ab such that a, b ∈ A(Ar) and the head of a coincides with the tail of b; uv such that u, v ∈ V (Ar) and u is adjacent to v; au(ua) such that a ∈ A(Ar) and u ∈ V (Ar) and the head (tail) of a is u; P a such that a ∈ A(Ar) and P ∈ P(Ar) and the arc a lies on the directed path P; PiPj such that Pi, Pj ∈ P(Ar) and it is possible to reach the head of Pj from the tail of Pi through a common vertex, and it is also possible to reach the head of Pi from the tail of Pj .In this paper, the concept of planarity of the directed pathos total digraph (that is, as an acyclic directed graph which can be drawn with non crossing arcs oriented in one direction) is being discussed and applied to a directed pathos total digraph of an arborescence Ar (DP T(Ar)). Further, the internal vertices of these directed pathos total digraph of Ar are cconsidered.Finally, the planarity of an arborescence resulting from the vertex-gluing of two directed paths ispresented and corresponding internal vertex number is obtained.

Polynomial bounds for chromatic number. III. Excluding a double star

AbstractA “double star” is a tree with two internal vertices. It is known that the Gyárfás–Sumner conjecture holds for double stars, that is, for every double star , there is a function such that if does not contain as an induced subgraph then (where are the chromatic number and the clique number of ). Here we prove that can be chosen to be a polynomial.

A simple linear time algorithm to solve the MIST problem on interval graphs

Motivated by the design of cost-efficient communication networks, the problem about Maximum Internal Spanning Tree that arises in a connected graph, is proposed to find a spanning tree with the maximum number of internal vertices. In 2018, Xingfu Li et al. presented a polynomial algorithm to find a maximum internal spanning tree in a connected interval graph. Based on the structure of normal orderings on interval graphs, we present a simple linear time algorithm that solve the problem when restricted to connected interval graphs in this paper. The proof provides additional insight about the linear time algorithm on interval graphs.

Analisis rainbow vertex connection pada beberapa graf khusus dan operasinya

The vertex colored graph G is said rainbow vertex cennected, if for every two vertices are connected by a path whose internal vertices have distinct colors. The rainbow vertex connection number of G, denoted by rvc(G), is the smallest number of colors that are needed in order to make G rainbow vertex connected. On this research, will be raised the issue of how to produce graphs the results of some special graph and how to find the rainbow vertex connection. Operation that use cartesian product, crown product, and shackle. Theorem in this research rainbow vertex connection number in graph the results of operations Wd3,m □ Pn,,Wd3,m ⵙ Pn, and shack(Btm,v,n).

Asymptotically sharpening the $s$-Hamiltonian index bound

For a non-negative integer $s\le |V(G)|-3$, a graph $G$ is $s$-Hamiltonian if the removal of any $k\le s$ vertices results in a Hamiltonian graph. Given a connected simple graph $G$ that is not isomorphic to a path, a cycle, or a $K_{1,3}$, let $\delta(G)$ denote the minimum degree of $G$, let $h_s(G)$ denote the smallest integer $i$ such that the iterated line graph $L^{i}(G)$ is $s$-Hamiltonian, and let $\ell(G)$ denote the length of the longest non-closed path $P$ in which all internal vertices have degree 2 such that $P$ is not both of length 2 and in a $K_3$. For a simple graph $G$, we establish better upper bounds for $h_s(G)$ as follows. \begin{equation*} h_s(G)\le \left\{ \begin{aligned} & \ell(G)+1, &&\mbox{ if }\delta(G)\le 2 \mbox{ and }s=0;\\ & \widetilde d(G)+2+\lceil \lg (s+1)\rceil, &&\mbox{ if }\delta(G)\le 2 \mbox{ and }s\ge 1;\\ & 2+\left\lceil\lg\frac{s+1}{\delta(G)-2}\right\rceil, && \mbox{ if } 3\le\delta(G)\le s+2;\\ & 2, &&{\rm otherwise}, \end{aligned} \right. \end{equation*} where $\widetilde d(G)$ is the smallest integer $i$ such that $\delta(L^i(G))\ge 3$. Consequently, when $s \ge 6$, this new upper bound for the $s$-hamiltonian index implies that $h_s(G) = o(\ell(G)+s+1)$ as $s \to \infty$. This sharpens the result, $h_s(G)\le\ell(G)+s+1$, obtained by Zhang et al. in [Discrete Math., 308 (2008) 4779-4785].

Better approximation algorithms for maximum weight internal spanning trees in cubic graphs and claw-free graphs

Given a connected vertex-weighted graph $G$, the maximum weight internal spanning tree (MaxwIST) problem asks for a spanning tree of $G$ that maximizes the total weight of internal nodes. This problem is NP-hard and APX-hard, with the currently best known approximation factor $1/2$ (Chen et al., Algorithmica 2019). For the case of claw-free graphs, Chen et al. present an involved approximation algorithm with approximation factor $7/12$. They asked whether it is possible to improve these ratios, in particular for claw-free graphs and cubic graphs. For cubic graphs we present an algorithm that computes a spanning tree whose total weight of internal vertices is at least $\frac{3}{4}-\frac{3}{n}$ times the total weight of all vertices, where $n$ is the number of vertices of $G$. This ratio is almost tight for large values of $n$. For claw-free graphs of degree at least three, we present an algorithm that computes a spanning tree whose total internal weight is at least $\frac{3}{5}-\frac{1}{n}$ times the total vertex weight. The degree constraint is necessary as this ratio may not be achievable if we allow vertices of degree less than three. With the above ratios, we immediately obtain better approximation algorithms with factors $\frac{3}{4}-\epsilon$ and $\frac{3}{5}-\epsilon$ for the MaxwIST problem in cubic graphs and claw-free graphs having no degree-2 vertices, for any $\epsilon>0$. The new algorithms are short (compared to that of Chen et al.) and fairly simple as they employ a variant of the depth-first search algorithm. Moreover, they take linear time while previous algorithms for similar problem instances are super-linear.

Truly non-trivial graphoidal graphs

A graphoidal cover of a graph G is a collection of non-trivial paths in G, which are not necessarily open, such that every vertex of G is an internal vertex of at most one path in and every edge of G is in exactly one path in If every path in a graphoidal of graph G has length at least 2, then graphoidal cover is called a truly non-trivial graphoidal cover (TNT graphoidal cover) of G. In this paper we obtain some structural necessary conditions for a graph to possess a TNT graphoidal cover. We also exhibit that these conditions are sufficient in the case of trees with diameter We also obtain a necessary condition in terms of order and size for a graph to possess a TNT graphoidal cover. Thereafter as an application of the necessary condition, we characterize complete graphs Kn and complete bipartite graphs possessing TNT graphoidal cover. Finally, we discuss regular graphs possessing TNT graphoidal cover and prove that every 3-regular graph possesses such a graphoidal cover.