We study the Equivalent Local Sequence Problem (ELSP), which consists in recovering an explicit sequence of local complementations that transforms a graph into a locally equivalent one. Focusing on directed Paley graphs, we establish that local complementations commute and induce a free action of an elementary abelian 2-group. The stabilizer condition is reformulated as a system of convolution equations and analyzed through Fourier techniques over finite fields, leading to a proof of stabilizer triviality. As a consequence, each graph in the orbit admits a unique subset encoding, and ELSP reduces to solving a linear inversion problem over 𝔽₂. This characterization completely resolves ELSP for directed Paley graphs, provides a polynomial-time inversion algorithm and highlights structural features that may support future developments in cryptographic frameworks and quantum graph-state models.

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.

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.

Let \(G\) be a connected graph. The center function defined on \(G\) yields a set of vertices that minimizes the maximum distance from the given input vertices. Through axiomatic characterization of the center function, we identify the specific axioms that characterize its behavior on connected graphs. Universal axioms encompass the properties satisfied by the center function on all connected graphs. However, for some graphs, the center function cannot be fully characterized using universal axioms alone. To address this, a set of graph class-specific axioms, known as non-universal axioms, was introduced. In the case of book graphs (Cartesian product of star graph \(K_{1,n}\) and path \(P_2\)), the center function cannot be adequately characterized using known universal axioms. Therefore, in this context, we find an axiomatic characterization of the center function on book graphs using the universal axioms and one newly introduced Cycle Consensus \((CC)\) axiom.

We study a search problem on capturing a moving target on an infinite real line. Two autonomous mobile robots (which can move with a maximum speed of 1) are initially placed at the origin, while an oblivious moving target is initially placed at distance d from the origin. The robots can move along the line in either direction, but the target is oblivious, cannot change direction, and moves either away from or toward the origin at a constant speed v. Our aim is to design efficient algorithms for two robots to capture the target. The target is captured only when both robots are co-located with it. The robots communicate only face-to-face (F2F), meaning they can exchange information only when co-located. We design algorithms under various knowledge scenarios regarding d, v, and the target’s direction of movement. We analyze competitive ratios, i.e., the capture time versus the optimal full-knowledge scenario, and show that our strategies use at most three direction changes.

We present a theorem which characterizes the class of line graphs of directed graphs. The characterization is an analogue of both the characterization of line graphs by Krausz (1943) and of directed line graphs of directed graphs by Harary and Norman (1960). Our characterization simplifies greatly in the case that the graph is bipartite. This and another result which we present draws attention to the special case of bipartite line graphs of directed graphs. As a result we explore the problem of finding the complete list of forbidden subgraphs for the class of bipartite line graphs of directed graphs. It appears, however, that this problem is extremely difficult. We do find two infinite families of forbidden subgraphs as well as several other illustrative examples.

A block graph is a graph in which each block (maximal biconnected subgraph) is a clique. A block graph can, equivalently, be seen as a tree of cliques where the blocks form complete subgraphs connected to each other in a tree-like, hierarchical structure. Such graphs provide a good conceptual and computational framework in designing, analyzing, and optimizing resilient and efficient power networks. In view of the above applications, this paper investigates the \(k\)-fault tolerant power domination number of block graphs. Also, we obtained a lower bound for \(k\)-fault tolerant power domination number when an edge in the graph is contracted.