The quadratic embedding (QE) class of the connected graphG= (V,E) is determined based on its quadratic embedding constant (QEC) of the distance matrix graphG. The distance matrix is a conditionally definite negative, or equivalently if it admits a quadratic embedding in a Hilbert space, or if QEC of the graphGis non-positive. In this study, the classification for the hairy cycle graphs such as bearded cycle graphBC_(k,m)forkis even, bearded cycle graphBC_(3,m)and broken sun graphBC_(4,m)will be calculated, and we derive the formulae of its QEC. These classes of graphs belongs to the QE class.

The connected graphG= (V,E) is classified as a Quadratic Embedding (QE) class based on the distance matrixDwhich is conditionally definite negative and if the quadratic embedding constant (QEC) of the graphGis non-positive. In this study, QEC be calculated for Squid Graph containingC₃,C₄,C₅, and kite graph containingC₄. All of these graphs are included in the QE class, withQEC= 0 for squid graphs and kite graphs containingC₄,QEC(Sq_(3,m)) < 0 for squid graphs containingC₃, and also for squid graphs containingC₅.

For a simple graph G, a vertex labeling f:V(G)\to {1, 2, ..., k} is called a k-labeling. The weight of a vertex v, denoted by $wt_f(v)$ is the sum of all vertex labels of vertices in the closed neighborhood of the vertex v. A vertex k-labeling is defined to be an inclusive distance vertex irregular distance k-labeling of G if for every two different vertices u and v there is $wt_f(u)\ne wt_f(v)$. The minimum k for which the graph G has a vertex irregular distance k-labeling is called the inclusive distance vertex irregularity strength of G. In this paper we establish a lower bound of the inclusive distance vertex irregularity strength for any graph and determine the exact value of this parameter for several families of graphs.

A modular irregular graph is a graph that admits a modular irregular labeling. A modular irregular labeling of a graph of order is a mapping of the set of edges of the graph to such that the weights of all vertices are different. The vertex weight is the sum of its incident edge labels, and all vertex weights are calculated with the sum modulo . The modular irregularity strength is the minimum largest edge label such that a modular irregular labeling can be done. In this paper, we construct a modular irregular labeling of two classes of graphs that are biregular; in this case, the regular double-star graph and friendship graph classes are chosen. Since the modular irregularity strength of the friendship graph also holds the minimal irregularity strength, then the labeling is also an irregular labeling with the same strength as the modular case.

An antimagic labeling of a graph G is a bijection $f:E(G)\to \{1,\dots ,|E(G)\left|\right\}$ such that the weights $w(x)={\sum }_{y\sim x}f(y)$ distinguish all vertices. A well-known conjecture of Hartsfield and Ringel (1990) is that every connected graph other than $K_2$ admits an antimagic labeling. For a set of distances D, a D-antimagic labeling of a graph G is a bijection $f:V(G)\to \{1,\dots ,|V(G)\left|\right\}$ such that the weight$\omega (x)={\sum }_{y\in N_D(x)}f(y)$ is distinct for each vertex x, where $N_D(x)=\{y\in V(G)|d(x,y)\in D\}$ is the D-neigbourhood set of a vertex x. If $N_D(x)=r$, for every vertex x in G, a graph G is said to be $(D,r)$-regular. In this paper, we conjecture that a graph admits a D-antimagic labeling if and only if it does not contain two vertices having the same D-neighborhood set. We also provide evidence that the conjecture is true. We present computational results that, for $D=\left\{1\right\}$, all graphs of order up to 8 concur with the conjecture. We prove that the set of $(D,r)$-regular D-antimagic graphs is closed under union. We provide examples of disjoint union of symmetric $(D,r)$-regular that are D-antimagic and examples of disjoint union of non-symmetric non-$(D,r)$-regular graphs that are D-antimagic. Furthermore, lastly, we show that it is possible to obtain a D-antimagic graph from a previously known distance antimagic graph.