Given a configuration of pebbles on the edges of a connected graph G, an edge pebbling move is defined as the removal of two pebbles off an edge and placing one on an adjacent edge. The domination cover edge pebbling number of a graph G is the minimum number of pebbles required such that the set of edges that contain pebbles form an edge dominating set S of G, for the initial configuration of pebbles can be altered by a sequence of pebbling moves and it is denoted by ψ_e(G) for a graph G. In this paper, we determine ψ_e(G) for Generalized Petersen graph, Jewel graph and Triangular snake graph.

In this paper we study group divisible designs (GDDs) with block size 4 and two groups of different sizes when λ₂ = 1. We obtain necessary conditions for the existence of such GDDs and prove that these necessary conditions are sufficient in several cases. Further, we present general constructions using resolvable designs.

In 1940, Birkhoff posed an open problem of counting all finite lattices on n elements. Recently, Bhavale counted all non-isomorphic lattices on n elements, containing up to four reducible elements, and having nullity up to three. Further, Aware and Bhavale counted all non-isomorphic lattices on n elements, containing up to five comparable reducible elements, and having nullity up to three. In this paper, we count all non-isomorphic lattices on n elements, containing five reducible elements, and having nullity three.

The unit graph of a commutative ring with a non-zero identity is a graph with vertices as ring elements, and there is an edge between two distinct vertices if their sum is a unit. This study investigates the decomposition of the unit graph by examining its induced subgraphs and analyze key graph invariants, such as connectivity, diameter, and girth, for a finite local ring. We further decompose the unit graph of certain finite commutative rings into fundamental structures, such as cycle and star graphs.

A mapping of the set of undirected simple (loopless) graphs to itself is a linear operator if it maps the edgeless graph to the edgeless graph and maps the union of graphs to the union of their images. A linear operator preserves a set if it maps that set to itself. We study linear operators that map sets defined by the restriction of their chromatic number. For example the set of all graphs whose chromatic number is at least \(k\) for some fixed \(3\leq k\leq n\). We show these linear operators must be vertex permutations.

The first Zagreb index of a graph \(G\) is defined as \(\sum\limits_{u \in V} d_G^2(u)\), where \(d_G(u)\) is the degree of vertex \(u\) in \(G\). The algebraic connectivity of a graph \(G\) is defined as the second smallest eigenvalue of the Laplacian matrix of \(G\). Using Wagner’s inequality, we in this paper first obtain an upper bound for the algebraic connectivity that involves the first Zagreb index of a graph. Following the ideas of obtaining the upper bound, we present sufficient conditions involving the first Zagreb index and the algebraic connectivity for some Hamiltonian properties of graphs.

We present a proof of a conjecture of Goh and Wildberger on the factorization of the spread polynomials. We indicate how the factors can be effectively calculated and exhibit a connection to the factorization of Fibonacci numbers into primitive parts.

For a graph \(G\), let \(la(G)\) denote the linear arboricity of \(G\) and \(\Delta(G)\) denote the maximum degree of \(G\). The famous linear arboricity conjecture was made by Akiyama, Exoo, and Harary [Covering and packing in graphs. IV. Linear arboricity] in 1981. It asserts that \(la(G) \leq \Bigl\lceil\frac{\Delta(G)+1}{2}\Bigr\rceil\). In this paper, we prove the linear arboricity conjecture for products of a path and a complete graph, and for products of a path and a tree.