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.

<p>For a graph <span class="math inline">\(G\)</span>, let <span class="math inline">\(la(G)\)</span> denote the linear arboricity of <span class="math inline">\(G\)</span> and <span class="math inline">\(\Delta(G)\)</span> denote the maximum degree of <span class="math inline">\(G\)</span>. 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 <span class="math inline">\(la(G) \leq \Bigl\lceil\frac{\Delta(G)+1}{2}\Bigr\rceil\)</span>. 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.</p>

<p>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 <span class="math inline">\(k\)</span> for some fixed <span class="math inline">\(3\leq k\leq n\)</span>. We show these linear operators must be vertex permutations.</p>

<p>The first Zagreb index of a graph <span class="math inline">\(G\)</span> is defined as <span class="math inline">\(\sum\limits_{u \in V} d_G^2(u)\)</span>, where <span class="math inline">\(d_G(u)\)</span> is the degree of vertex <span class="math inline">\(u\)</span> in <span class="math inline">\(G\)</span>. The algebraic connectivity of a graph <span class="math inline">\(G\)</span> is defined as the second smallest eigenvalue of the Laplacian matrix of <span class="math inline">\(G\)</span>. 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.</p>

<p>Let <span class="math inline">\(G\)</span> be a connected graph. The center function defined on <span class="math inline">\(G\)</span> 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 <span class="math inline">\(K_{1,n}\)</span> and path <span class="math inline">\(P_2\)</span>), 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 <span class="math inline">\((CC)\)</span> axiom.</p>

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.

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.

<p>Let <span class="math inline">\(G\)</span> be a graph. We introduce the <span>balanced antimagic labeling</span> as an analogue to the antimagic labeling. A <span><span class="math inline">\(k\)</span>-total balanced antimagic</span> labelling is a map <span class="math inline">\(c\colon V (G)\cup E(G) \to \{1,2,\ldots,k\}\)</span> such that: the label classes differ in size by at most one, each vertex <span class="math inline">\(x\)</span> is assigned the weight <span class="math inline">\(w(x)={c}(x)+\sum\limits_{x\in e}{c}(e)\)</span>, <span>and</span> <span class="math inline">\(w(x)\neq w(y)\)</span> for <span class="math inline">\(x\neq y\)</span>.</p> <p>We present several properties of balanced antimagic labeling. We also derive such a labeling for complete graphs and complete bipartite graphs.</p>

<p>Exploring the vulnerability of any real-life network helps designers understand how strongly components or elements of the network are connected and how well they can function if there is any disruption. Any chemical structure can also be considered as a network in which the atoms correspond to the vertices, and the chemical bonds between the atoms correspond to the edges. Let <span class="math inline">\(G=(V, E)\)</span> represent any simple graph with vertex set <span class="math inline">\(V\)</span> and edge set <span class="math inline">\(E\)</span>. The vulnerability measure used in this paper is the paired domination integrity, defined as the minimum of the sum of any paired dominating set <span class="math inline">\(S\)</span> of a graph <span class="math inline">\(G\)</span> and the order of the largest component in the induced subgraph of <span class="math inline">\(V-S\)</span>. The minimum is found by considering all possible paired dominating sets of <span class="math inline">\(G\)</span>. In this paper, we obtain the paired domination integrity of the comb product of paths and cycles. In addition, we extend the study of graph vulnerability to chemical structures.</p>

<p>Given a network modeled as a graph, a detection system is a subset of vertices equipped with “detectors” that can uniquely identify an “intruder” anywhere in the graph. We consider two types of detection systems: open-locating-dominating (OLD) sets and identifying codes (ICs). In an OLD set, each vertex has a unique, non-empty set of detectors in its open neighborhood; meanwhile, in an IC, each vertex has a unique, non-empty set of detectors in its closed neighborhood. We explore one of their fault-tolerant variants: redundant OLD (RED:OLD) sets and redundant ICs (RED:ICs), which ensure that removing/disabling at most one detector retains the properties of OLD sets and ICs, respectively. This paper focuses on constructing optimal RED:OLD sets and RED:ICs on the infinite king grid, and presents the proof for the bounds on their minimum densities; <span class="math inline">\(\left[\frac{3}{10}, \frac{1}{3}\right]\)</span> for RED:OLD sets and <span class="math inline">\(\left[\frac{3}{11}, \frac{1}{3}\right]\)</span> for RED:ICs.</p>