An edge-magic total \((EMT)\) labeling on a graph \(G\) is
a one-to-one mapping \(\lambda : V(G) \cup E(G) \to {1,2,—,|V(G)| +
|E(G)|}\) such that the set of edge weights is one point set, i.e. for
any edge \(xy \in G, w(xy) = {a}\) where \(a = \lambda(x) + \lambda(y) + \lambda(xy)\)
is called a magic constant. If \(\lambda(V(G)) = {1,2,—,|V(G|}\) then an
edge-magic total labeling is called a super edge-magic total labeling.
In this paper, we formulate a super edge-magic total labeling for
a particular tree family called subdivided star \(T(l_1,l_2,\ldots,l_p)\) for
\(p>3\).

Let \(G\) be an edge-colored graphs. A heterochromatic path of \(G\) is such a path in which no two edges have the same color. Let \(g^c(G)\) and \(d^c(v)\) denote the heterochromatic girth and the color degree of a vertex \(v\) of \(G\), respectively. In this paper, some color degree and heterochromatic girth conditions for the existence of heterochromatic paths are obtained.

Let \(\mathcal{U}_m^{W}\) denote the set of unicyclic weighted graphs of size \(m\) with weight \(W\). In this paper, we determine the weighted graph in \(\mathcal{U}_m^{W}\) with maximum spectral radius.

A subset of vertices of a graph \(G\) is called a feedback vertex set of \(G\) if its removal results in an acyclic subgraph. In this paper, we investigate the feedback vertex set of generalized Kautz digraphs \(GK(2,n)\). Let \(f(2,n)\) denote the minimum cardinality over all feedback vertex sets of the generalized Kautz digraph \(GK(2,n)\). We obtain the upper bound of \(f(2,n)\) as follows:
\[f(2,n) \leq n-(\left\lfloor \frac{n}{3} \right\rfloor + \left\lfloor \frac{{n-2}}{3} \right\rfloor + \lfloor \frac{n-8}{9}\rfloor)\].

Let \(G\) be a graph of order \(n\) and let \(\mu\) be an eigenvalue of multiplicity \(m\). A star complement for \(\mu\) in \(G\) is an induced subgraph of \(G\) of order \(n-m\) with no eigenvalue \(\mu\). In this paper, we investigate maximal and regular graphs that have \(K_{r,s} + t{K_{1}}\) as a star complement for \(\mu\) as the second largest eigenvalue. Interestingly, it turns out that some well-known strongly regular graphs are uniquely determined by such a star complement.

Given a graph \(G\) and a non-negative integer \(g\), the \(g\)-extra-connectivity of \(G\), denoted by \(\kappa_g(G)\), is the minimum cardinality of a set of vertices of \(G\), if any, whose deletion disconnects \(G\) and every remaining component has more than \(g\) vertices. Note that \(\kappa_0(G)\) and \(\kappa_1(G)\) correspond to the usual connectivity and restricted vertex connectivity of \(G\), respectively. In this paper, we determine \(\kappa_g(FQ_n)\) for \(0 \leq g \leq n-4\), \(n \geq 8\), where \(FQ_n\) denotes the \(n\)-dimensional folded hypercube.

The construction of association schemes based on the subspaces of type \((2,0,1)\) in singular symplectic space over finite fields is provided in this paper.Applying the matrix method and combinatorial design theory, all parameters of the association scheme are computed.

In this paper we define new types of generalizations in the distance sense of Lucas numbers. These generalizations are based on introduced recently the concept of \((2, k)\)-distance Fibonacci numbers.We study some properties of these numbers and present identities
which generalize known identities for Lucas numbers. Moreover, we show representations and interpretations of these numbers.

The number of colors required to properly color the edges of a simple graph \(G\) such that any two vertices are incident with different sets of colors is referred to as the vertex-distinguishing edge chromatic number, denoted by \(\chi’_{vd}(G)\). This paper explores an interesting phenomenon concerning vertex-distinguishing proper edge coloring. Specifically, we prove that for every integer \(m \geq 3\), there exists a graph \(G\) of maximum degree \(m\) with \(\chi’_{vd}(G) < \chi'_{vd}(H)\), where \(H\) is a proper subgraph of \(G\).

Let \(P\) be a planar point set with no three points collinear. A \(k\)-hole of \(P\) is a convex \(k\)-gon \(H\) such that the vertices of \(H\) are elements of \(P\) and no element of \(P\) lies inside \(H\). In this article, we prove that for any planar \(9\)-point set \(P\) with no three points collinear and at least \(5\) vertices on the boundary of the convex hull, \(P\) contains a \(5\)-hole and a disjoint \(3\)-hole.