We introduce and investigate a new class of graph maps, called backward edge-preserving maps. These are the maps between the vertex sets of graphs such that the pre-image of every edge in the target graph is an edge in the source graph. We give a complete structural characterization of such maps for both general and connected graphs. We also explore the relationship between backward edge-preserving maps, metric maps, and graph homomorphisms, and establish criteria for the intersection of these classes. Furthermore, we study a related class of graph cohomomorphisms and establish conditions under which a map can simultaneously be a homomorphism and a cohomomorphism.

A directed star is a star digraph in which the centre is either a sink or a source. With every directed star, we associate a weight \(w\left(\alpha\right)\). Let \(D\) be a digraph; and \(C\), a spanning subgraph of \(D\); in which every component is a directed star. Then \(C\) is called a directed star cover of \(D\), and the weight of \(C\) is \(w(C)=\prod_{\alpha}w\left(\alpha\right)\), where the product is taken over all the components \(\alpha\) of \(C\). The directed star polynomial of \(D\) is \[E\left(D;\mathbf{w}\right)=\sum w(C),\] where the sum is taken over all the directed star covers of \(D\). The paper establishes properties of star polynomials and establishes relationships between star polynomials, source polynomials (where all components are source stars), and sink polynomials (where all components are sink stars). Furthermore, it presents methods to compute these polynomials for general digraphs. We derive formulae for the star polynomials of rooted products of digraphs. Moreover, we establish applications of these polynomials to several domination parameters and obtain results regarding these polynomials and independent sets in undirected graphs.

Let \(G\) be an undirected graph. The \(k\)-strong labeling (coloring) number of \(G\), \(\chi_k(G)\), is the minimum number of labels (colors) necessary to label all the vertices of \(G\) so that no two vertices that are exactly of distance \(k\) apart have the same label. \(\chi_1\) is the usual chromatic number. In this article it is shown that any linear operator on the set of graphs on \(n\) vertices thar preserves \(\chi_k\) is a vertex permutation.

Cartesian-product networks combine well-studied graphs to create new structures with inherited properties, making them valuable for interconnection networks and parallel algorithms. Cycle decompositions of these networks are crucial for fault tolerance and adaptive routing. In this paper, we address the hypercube version of the Oberwolfach problem, focusing on decompositions of \(Q_n\) into cycles of equal or unequal lengths. We present an algorithm that enumerates all possible cycle types in \(Q_n\) and determine which decompositions are feasible or infeasible for \(Q_4\). Using an inductive approach, we extend these results to \(Q_n\) by leveraging distinct perfect matchings of \(Q_4\), yielding a variety of cycle decompositions. Additionally, we present results on factorizations of \(Q_n\) when \(n\) is a power of \(2\). These findings enhance the understanding of cycle structures in hypercubes and their applications to interconnection networks.

We investigate the combinatorial structure of edge-disjoint triangle packings in the complete graph \(K_n\). Two triangles are said to be edge-disjoint if they share no common edges, though they may share at most one vertex. For a given \(n\), let \(T_n\) denote the total number of subsets of triangles in \(K_n\) that are pairwise edge-disjoint, including the empty set, and let \(T_n^k\) denote the number of \(k\)-element sets of such triangles. In this article, we establish: (i) a general recurrence relation for \(T_n\) that enables asymptotic analysis, yielding the growth \(\log T_n = \Theta(n^2 \log n)\) for large \(n\); (ii) exact closed-form formulas for the number of edge-disjoint pairs (\(T_n^2\)), triples (\(T_n^3\)), and quadruples (\(T_n^4\)) of triangles in \(K_n\) for \(n \geq 6\). These results extend classical work on Steiner Triple Systems and provide new tools for analyzing triangle packings in complete graphs.

This paper introduces Hadamard-type \(t\)-Fibonacci-Lehmer (HTFL) sequences, a new hybrid construction combining Lehmer and Fibonacci recurrences. We establish their fundamental properties, including simple periodicity, and extend the definition to finite groups, with a detailed study of the Heisenberg group. Building on these results, we propose two Diffie–Hellman-style key exchange protocols based on upper-triangular unipotent matrices parameterized by HTFL sequence terms. Our work thus connects sequence theory, group theory, and cryptography in a novel way. While the algebraic framework and periodicity analysis are rigorous, we present the cryptographic constructions primarily as a conceptual foundation. We also discuss potential security considerations and outline directions for strengthening these schemes under formal hardness assumptions. This study demonstrates that HTFL sequences provide a fertile ground for both combinatorial investigations and future cryptographic applications.

A split graph is a graph in which the vertices can be partitioned into an independent set and a clique. We show that every nonsplit graph has at most four split maximal proper edge induced subgraphs. The exhaustive list of fifteen classes of nonsplit graphs having a split maximal proper edge induced subgraph is determined in this paper.

A kernel \(J\) of a digraph \(D\) is an independent set of vertices of \(D\) such that for every \(z\in V(D)\backslash J\) there exists an arc from \(z\) to \(J.\) A digraph \(D\) is said to be kernel-perfect if every induced subdigraph of it has a kernel. We characterise kernel-perfectness in special families of digraphs, namely, the line digraph, the subdivision digraph, the middle digraph, the digraph \(R(D)\) and the total digraph. We also obtain some results on kernel-perfectness in the generalised Mycielskian of digraphs. Moreover, we find some new classes of kernel-perfect digraphs by introducing a new product on digraphs.