The Wiener polarity index of a graph \(G\), denoted by \(W_p(G)\), is the number of unordered pairs of vertices \(u, v\) such that the distance between \(u\) and \(v\) is three, introduced by Harold Wiener in 1947. This index is utilized to demonstrate quantitative structure-property relationships in various acyclic and cyclic hydrocarbons. In this paper, we investigate the Wiener polarity index on the Cartesian, direct, strong, and lexicographic products of two non-trivial connected graphs.

Given a graph \(G := (V, E)\) and an integer \(k \geq 2\), the \({component \;order\; edge connectivity}\) of \(G\) is the smallest size of an edge set \(D\) such that the subgraph induced by \(G – D\) has all components of order less than \(k\). Let \({G}(n,m)\) denote the collection of simple graphs \(G\) with \(n\) vertices and \(m\) edges. In this paper, we investigate properties of component order edge connectivity for \({G}(n,m)\), particularly proving results on the maximum and minimum values of this connectivity measure for \({G}(n,m)\) specific values of \(n\), \(m\), and \(k\).

Let \(D\) be a simple digraph without loops and parallel arcs. Deng and Kelmans [A. Deng, A. Kelmans, Spectra of digraph transformations, Linear Algebra and its Applications, \(439(2013) 106-132]\) gave the definition of transformation digraphs by introducing symbol \(‘0’\) and \(‘1’\), and investigated the regular and spectra of digraph transformation. In this paper we discuss a class of total transformation digraphs associate with symbol \(‘0’\). Furthermore, we determine the regularity of these ten new kinds of total transformation digraphs and also give necessary and sufficient conditions for them to be strongly connected.

In this paper, using the generating function, we derive Binet formulas and determinant expressions for the k-generalized Fibonacci numbers and Lucas numbers. As applications, we obtain some new recurrence relations for the Stirling numbers of the second kind and power sums.

A graph is said to be symmetric if its automorphism group acts transitively on its arcs. Let \(p\) be a prime. In [J. Combin. Theory B \(97 (2007) 627-646]\), Feng and Kwak classified connected cubic symmetric graphs of order \(4p\) or \(4p^2\). In this article, all connected cubic symmetric graphs of order \(4p^2\) are classified. It is shown that up to isomorphism there is one and only one connected cubic symmetric graph of order \(4p^3\) for each prime \(p\), and all such graphs are normal Cayley graphs on some groups.

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.