A zero divisor graph on a finite commutative ring \(\mathfrak{R}\) is a graph with set of vertices consists of zero divisor elements \(Z(\mathfrak{R})\) of the ring, and we have an edge between any two elements in \(\mathfrak{R}\) if their product is the zero element. In this work, we will explore some zero divisor graph invariants constructed on the rings of the form \(\mathfrak{R}=\mathbb{Z}_n,\) when \(n\) is a product of square free primes. In particular, we will find the radio number for zero-divisor graphs constructed on \(\mathbb{Z}_{\mathfrak{p}_1 \mathfrak{p}_2 \mathfrak{p}_3}\) where \(\mathfrak{p}_1, \mathfrak{p}_2,\) and \(\mathfrak{p}_3\) are distinct primes with \(2 \leq \mathfrak{p}_3 < \mathfrak{p}_2 < \mathfrak{p}_1\) and combining with the known results for \(\mathbb{Z}_{\mathfrak{p}^3}\) and \(\mathbb{Z}_{\mathfrak{p}_1^2 \mathfrak{p}_2}\), It covers all possible cases when \(n\) is divisible by at most three primes.

A graph \(G(V, E)\) is word-representable if there exists a word \(w\) over the alphabet \(V\) such that for distinct letters \(x,y\in V\), \(x\) and \(y\) alternate in \(w\) if and only if they are adjacent in \(G\). In general, determining whether a graph is word-representable is an NP-complete problem. A graph is co-bipartite if its complement is bipartite. Therefore, the vertex set of a co-bipartite graph can be partitioned into two disjoint subsets \(X\) and \(Y\) such that the subgraphs induced by \(X\) and \(Y\) are cliques. Necessary and sufficient conditions for a co-bipartite graph to be word-representable in terms of a vertex ordering are known. Based on this ordering, we study the representation number of word-representable co-bipartite graphs and analyse the speed and entropy of this graph class. We show that the representation number of any word-representable co-bipartite graph is at most \(3\), and permutation graphs are the only co-bipartite graphs with representation number \(2\). We prove that the speed is at most \(2^{O(n \log n)}\) and the entropy is \(0\). In particular, we obtain an upper bound on the number of labelled graphs in this class, which is significantly smaller than the known bound for the class of all co-bipartite graphs. These results provide a better understanding of the structure and enumeration of word-representable co-bipartite graphs and show that vertex ordering is an effective tool for studying this class.

Spread of information within a wide variety of systems can be represented as evolving processes on digraphs. Starting from a random subset of initially active vertices, the measures we introduce assess the probability, speed, or number of steps it takes to spread information to the entire digraph, thus achieving digraph synchrony. Some of these measures may be viewed as generalizations of digraph connectivity or as generalizations of the diameter of a digraph to higher-order diameters. The paper places considerable emphasis on the regular case of Cayley digraphs associated to finite groups. It is demonstrated that, with appropriate assumptions on the growth of the generating sets, all the higher-order diameters of random Cayley digraphs are almost surely at most 2, as the digraph order goes to infinity. Certain results on the velocity of spread of information in digraphs are also presented.

We introduce and study a new family of circulant digraphs associated with the cyclic group \({\mathbb Z}_N\), obtained by restricting admissible combinations of two generators \(a\) and \(b\) to three coordinate sectors. The resulting distance-like function differs from the standard directed distance in circulant digraphs and gives rise to new geometric and combinatorial phenomena. Using planar lattice representations and periodic tessellations, we analyze the growth of reachable sets and derive Moore-type upper bounds for the corresponding order/diameter problem. We construct explicit infinite families of three-quarters circulant structures with the prescribed diameter and provide lattice-based methods for determining admissible generator pairs. Separate constructions are obtained for even and odd diameters. In addition, computational experiments for small and moderate orders suggest improved families for even diameters and motivate a conjectural asymptotic formula for the maximum attainable order. The paper highlights the interplay between constrained lattice representations, periodic tilings, and extremal problems for circulant networks.

Let \(G\) be a simple connected graph and \(A(G)\) and \(D(G)\) represent the adjacency matrix and the diagonal matrix of degrees of graph G, respectively. The normalized Laplacian of \(G\) is defined by \(\mathcal{L}(G)=I_n-D(G)^{-1/2}A(G)D(G)^{-1/2},\) where \(I_n\) is the identity matrix of order \(n\). The normalized Laplacian plays an important role in spectral graph theory. In this paper, we characterize the connected graphs that minimize the spectral radius of the normalized Laplacian in the class of graphs with exactly one vertex of degree greater than two and in the class of graphs with exactly two vertices of degree greater than two. In each class, we determine the extremal graphs and the exact minimum normalized Laplacian spectral radius.

The inequality chain \(ir(G)\le \gamma(G)\le i(G)\le \alpha(G) \le \Gamma(G) \le I\!R(G)\) is known as the domination chain, where \(ir(G), \gamma(G), i(G), \alpha(G), \Gamma(G)\) and \(I\!R(G)\) are the lower irredundance number, the domination number, the independence domination number, the independence number, the upper domination number and the upper irredundance number of \(G\), respectively. The Ramsey-type problem seeks to characterize the family \({\mathcal H}\) of graphs such that every \({\mathcal H}\)-free graph \(G\) has a bounded parameter \(\mu\). The classical Ramsey’s theorem states that every \(\{K_n, E_n\}\)-free graph has a bounded number of vertices. Furuya (Discrete Math.Theor 2018) characterized \({\mathcal H}\) such that every connected \({\mathcal H}\)-free graph \(G\) has a bounded domination number. The characterization of the graph family \({\mathcal H}\) for which every connected \({\mathcal H}\)-free graph \(G\) has a bounded independence number was due to Choi, Furuya, Kim, Park (Discrete math. 2020) and Chiba, Furuya (Electron. J. Combin., 2022). In this paper, we further characterize \({\mathcal H}\) such that every connected \({\mathcal H}\)-free graph \(G\) has bounded \(\mu(G)\) for \(\mu\) belonging to the set \(\{ir(G), i(G), \Gamma(G), \text{IR}(G)\}\). This completes the characterization of \({\mathcal H}\) for which every connected \({\mathcal H}\)-free graph \(G\) has bounded \(\mu(G)\) for \(\mu(G)\) along the domination chain. Additionally, we characterize \({\mathcal H}\) such that every connected \({\mathcal H}\)-free graph \(G\) has bounded \(\mu(G)\) for \(\mu\) related to the domination number. Specifically, we consider the following parameters of \(G\): open irredundance number \(O\!I\!R(G)\), independence saturation number \(I\!S(G)\) and irredundance saturation number \(I\!R\!S(G)\).

Let \(D\) be a finite simple digraph with vertex set \(V(D)\). For \(v\in V(D)\), the set \(N^-[v]\) consists of \(v\) and all vertices of \(D\) from which arcs go into \(v\). Let \(k\ge 1\) be an integer. A signed double Roman \(k\)-dominating function (SDR\(k\)DF) on a digraph \(D\) is a function \(f:V(D)\rightarrow\{-1,1,2,3\}\) satisfying the following conditions: (i) \(\sum\limits_{x\in N^-[v]}f(x)\ge k\) for each \(v\in V(D)\); (ii) every vertex \(u\) with \(f(u)=-1\) has an in-neighbor \(z\) with \(f(z)=3\) or two in-neighbors \(x\) and \(y\) with \(f(x)=f(y)=2\); (iii) every vertex \(u\) with \(f(u)=1\) has an in-neighbor \(z\) with \(f(z)\ge 2\). The weight of an SDR\(k\)DF \(f\) is \(\omega(f)=\sum\limits_{v\in V(D)}f(v)\). The signed double Roman \(k\)-domination number \(\gamma_{sdR}^k(D)\) is the minimum weight of an SDR\(k\)DF on \(D\). In this paper, we study the signed double Roman \(k\)-domination number of digraphs and present various bounds on \(\gamma_{sdR}^k(D)\). In addition, we determine this parameter for several classes of digraphs. Some of our results extend well-known properties of the signed double Roman \(k\)-domination number \(\gamma_{sdR} ^k(G)\) of graphs \(G\).

We give combinatorial interpretations of some Rogers\(-\)Ramanujan type identities, also known as sum-product identities in terms of \((n+t)-\)color partitions and split \((n+t)-\)color partitions. The identities discussed in this study contains negative exponent of \(q\). These interesting results reveal rich structure and great potential for further research because they reveal intricate mathematical structures, and link various other fields.

Let \(G\) be a graph of a network system with vertices, \(V(G)\), representing physical locations and edges, \(E(G)\), representing informational connectivity. A locating-dominating (LD) set \(S \subseteq V(G)\) is a subset of vertices representing detectors capable of sensing an “intruder” at precisely their location or at some unknown point in their open-neighborhood. An LD set must be capable of locating an intruder anywhere in the graph using this collection of detectors. We explore three types of fault-tolerant LD sets: redundant LD sets, which allow at most one detector to be removed or disabled, error-detecting LD sets, which allow at most one false negative, and error-correcting LD sets, which allow at most one error (false positive or false negative). In particular, we determine lower and upper bounds for the minimum density of these three fault-tolerant locating-dominating sets in the infinite king grid.

Several necessary properties of König–Egerváry graphs involving the core, the corona, and critical independent sets are by now part of the folklore of the theory, and have motivated different lines of research within the same framework. In particular, every König–Egerváry graph satisfies the core–corona identity \(|core(G)|+|corona(G)|=2\alpha(G),\) the covering relation \(corona(G)cup N(core(G))=V(G),\) and the fact that \(core(G)\) is a critical independent set. Each of these conditions captures a different aspect of the interaction between maximum independent sets and matchings, but none of them alone characterizes the König–Egerváry property. In this note we show that their conjunction does: a graph \(G\) is König–Egerváry if and only if the above two core–corona conditions hold and \(core(G)\) is critical. Equivalently, the class of König–Egerváry graphs is precisely the intersection of the three graph families determined by these conditions. We also provide examples showing that the characterization is sharp: any two of the three conditions may hold in a graph which is not König–Egerváry.