Solution of the random field XY magnet on a fully connected graph

We use large deviation theory to obtain the free energy of the XY model on a fully connected graph on each site of which there is a randomly oriented field of magnitude $h$. The phase diagram is obtained for two symmetric distributions of the random orientations: (a) a uniform distribution and (b) a distribution with cubic symmetry. In both cases, the ordered state reflects the symmetry of the underlying disorder distribution. The phase boundary has a multicritical point which separates a locus of continuous transitions (for small values of $h$) from a locus of first order transitions (for large $h$). The free energy is a function of a single variable in case (a) and a function of two variables in case (b), leading to different characters of the multicritical points in the two cases.

Graph Realizations Constrained by Connected Local Dimensions and Connected Local Bases

For an ordered set W = {w1,w2, ...,wk} of k distinct vertices in a connected graph G, the representation of a vertex v of G with respect to W is the k-vector r(v|W) = (d(v,w1), d(v,w2), ..., d(v,wk)), where d(v,wi) is the distance from v to wi for 1 ≤ i ≤ k. The setW is called a connected local resolving set of G if the representations of every two adjacent vertices of G with respect to W are distinct and the subgraph ⟨W⟩ induced by W is connected. A connected local resolving set of G of minimum cardinality is a connected local basis of G. The connected local dimension cld(G) of G is the cardinality of a connected local basis of G. In this paper, the connected local dimensions of some well-known graphs are determined. We study the relationship between connected local bases and local bases in a connected graph, and also present some realization results.

Fault-Tolerant Metric Dimension of Circulant Graphs

Let G be a connected graph with vertex set V(G) and d(u,v) be the distance between the vertices u and v. A set of vertices S={s1,s2,…,sk}⊂V(G) is called a resolving set for G if, for any two distinct vertices u,v∈V(G), there is a vertex si∈S such that d(u,si)≠d(v,si). A resolving set S for G is fault-tolerant if S\{x} is also a resolving set, for each x in S, and the fault-tolerant metric dimension of G, denoted by β′(G), is the minimum cardinality of such a set. The paper of Basak et al. on fault-tolerant metric dimension of circulant graphs Cn(1,2,3) has determined the exact value of β′(Cn(1,2,3)). In this article, we extend the results of Basak et al. to the graph Cn(1,2,3,4) and obtain the exact value of β′(Cn(1,2,3,4)) for all n≥22.

Hamiltonian Laceability in Book Graphs

A connected graph network G is a Hamilton-t-laceable (Hamilton- t*-laceable) if Ǝ in G a Hamilton path connecting each pair (at least one pair) of vertices at distance ‘t’ in G such that 1 ≤ t ≤ diameter (G). In this paper, we discuss the f-edge-Hamilton-t-laceability of Book and stacked Book graphs, where 1 ≤ t ≤ diameter (G).

Facial Expressions Recognition Based on Delaunay Triangulation of Landmark and Machine Learning

Facial expressions can tell a lot about an individual’s emotional state. Recent technological advances opening avenues for automatic Facial Expression Recognition (FER) based on machine learning techniques. Many works have been done on FER for the classification of facial expressions. However, the applicability to more naturalistic facial expressions remains unclear. This paper intends to develop a machine learning approach based on the Delaunay triangulation to extract the relevant facial features allowing classifying facial expressions. Initially, from the given facial image, a set of discriminative landmarks are extracted. Along with this, a minimal landmark connected graph is also extracted. Thereby, from the connected graph, the expression is represented by a one-dimensional feature vector. Finally, the obtained vector is subject for classification by six well-known classifiers (KNN, NB, DT, QDA, RF and SVM). The experiments are conducted on four standard databases (CK+, KDEF, JAFFE and MUG) to evaluate the performance of the proposed approach and find out which classifier is better suited to our system. The QDA approach based on the Delaunay triangulation has a high accuracy of 96.94% since it only supports non-zero pixels, which increases the recognition rate.

The Partition Dimension of a Path Graph

Resolving partition is part of graph theory. This article, explains about resolving partition of the path graph, with. Given a connected graph and is a subset of writen . Suppose there is , then the distance between and is denoted in the form . There is an ordered set of -partitions of, writen then the representation of with respect tois the The set of partitions ofis called a resolving partition if the representation of each to is different. The minimum cardinality of the solving-partition to is called the partition dimension of G which is denoted by . Before getting the partition dimension of a path graph, the first step is to look for resolving partition of the graph. Some resolving partitions of path graph, with , and are obtained. Then, the partition dimension of the path graph which is the minimum cardinality of resolving partition, namely pd (Pn)=2Resolving partition is part of graph theory. This article, explains about resolving partition of the path graph, with. Given a connected graph and is a subset of writen . Suppose there is , then the distance between and is denoted in the form . There is an ordered set of -partitions of, writen then the representation of with respect tois the The set of partitions ofis called a resolving partition if the representation of each to is different. The minimum cardinality of the solving-partition to is called the partition dimension of G which is denoted by . Before getting the partition dimension of a path graph, the first step is to look for resolving partition of the graph. Some resolving partitionsof path graph, with , and are obtained. Then, the partition dimension of the path graph which is the minimum cardinality of resolving partition, namely.

Vertex Identification in Grids and Prisms

A red-white coloring of a nontrivial connected graph [Formula: see text] of diameter [Formula: see text] is an assignment of red and white colors to the vertices of [Formula: see text] where at least one vertex is colored red. Associated with each vertex [Formula: see text] of [Formula: see text] is a [Formula: see text]-vector, called the code of [Formula: see text], whose [Formula: see text]th coordinate is the number of red vertices at distance [Formula: see text] from [Formula: see text]. A red-white coloring of [Formula: see text] for which distinct vertices have distinct codes is called an identification coloring or ID-coloring of [Formula: see text]. A graph [Formula: see text] possessing an ID-coloring is an ID-graph. The minimum number of red vertices among all ID-colorings of an ID-graph [Formula: see text] is the identification number or ID-number of [Formula: see text]. It is shown that the grid [Formula: see text] is an ID-graph if and only [Formula: see text] and the prism [Formula: see text] is an ID-graph if and only if [Formula: see text].

On local antimagic vertex coloring of corona products related to friendship and fan graph

Let <em>G</em>=(<em>V</em>,<em>E</em>) be connected graph. A bijection <em>f </em>: <em>E</em> → {1,2,3,..., |<em>E</em>|} is a local antimagic of <em>G</em> if any adjacent vertices <em>u,v</em> ∈ <em>V</em> satisfies <em>w</em>(<em>u</em>)≠ <em>w</em>(<em>v</em>), where <em>w</em>(<em>u</em>)=∑_e∈E(u)<em>f</em>(<em>e</em>), <em>E</em>(<em>u</em>) is the set of edges incident to <em>u</em>. When vertex <em>u</em> is assigned the color <em>w</em>(<em>u</em>), we called it a local antimagic vertex coloring of <em>G</em>. A local antimagic chromatic number of <em>G</em>, denoted by <em>χ</em>_la(<em>G</em>), is the minimum number of colors taken over all colorings induced by the local antimagic labeling of <em>G</em>. In this paper, we determine the local antimagic chromatic number of corona product of friendship and fan with null graph on <em>m</em> vertices, namely, <em>χ</em>_la(<em>F</em>_n ⊙ \overline{K_m}) and <em>χ</em>_la(<em>f</em>_(1,n) ⊙ \overline{K_m}).