Laplacian polynomial of a graph product and its application

Let be a connected noncomplete graph with at least 3 vertices. Then an independent set that dominates , in which every vertex of is at distance exactly two from another vertex in , is called an independent semitotal dominating set (ISTd-set) of . The independent semitotal domination number, denoted by , is the minimum cardinality of such a set. This paper investigates the Cartesian product , tensor product , and strong product of graphs, where is a connected noncomplete graph and is a nontrivial complete graph. Furthermore, the independent semitotal domination number for each of these graph products is determined.

ABSTRACT The connectivity of a network is directly related to its fault tolerance. The H -structure connectivity is a generalization of the classical connectivity of a graph. In this paper, we give a tight upper bound on the H -structure connectivity of a graph and obtain the precise values of the path-structure and star-structure connectivity of complete bipartite graphs. For Cartesian product graphs, lexicographic product graphs and complete product graphs, the relationship between the structure connectivity of the product graphs and the structure connectivity of the factor graphs is explored. These results are more universal and make the contents of the structure connectivity richer in terms of product graphs.

In this paper, we study the total co-independent domination (TC-ID) number of various classes of graphs and their Cartesian products. We determine the TC-ID number for Cartesian products of complete graphs taken finitely many times, as well as for the Cartesian product of a complete graph with an arbitrary connected graph. Furthermore, we obtain additional results on the TC-ID number.

Let $\mathcal{F}$ be a set of subsets of a set $W$. When is there a tree $T$ with vertex set $W$ such that each member of $\mathcal{F}$ is the set of vertices of a subtree of $T$? It is necessary that $\mathcal{F}$ has the Helly property and the intersection graph of $\mathcal{F}$ is chordal. We will show that these two necessary conditions are together sufficient in the finite case, and more generally, they are sufficient if no element of $W$ belongs to infinitely many infinite sets in $\mathcal{F}$.

On g-extra connectivity of corona-type graph products

Intersections of Graphs and \({\chi }\)-Boundedness

Abstract In the Maker–Breaker resolving game, two players named Resolver and Spoiler alternately select unplayed vertices of a given graph G . The aim of Resolver is to select all the vertices of some resolving set of G , while Spoiler aims to select at least one vertex from every resolving set of G . In this paper, this game is investigated on the lexicographic product of graphs. It is proved that if Spoiler has a winning strategy on a graph H no matter who starts the game, or if the first player has a winning strategy on H , then Spoiler always has a winning strategy on $$G\circ H$$ G ∘ H . Special attention is paid to lexicographic products in which the second factor is a complete graph, a path, or a cycle. For instance, in $$G\circ P_{2\ell }$$ G ∘ P 2 ℓ and in $$G\circ C_{2\ell }$$ G ∘ C 2 ℓ , Resolver always wins, while in $$G\circ P_{2\ell +1}$$ G ∘ P 2 ℓ + 1 and in $$G\circ C_{2\ell +1}$$ G ∘ C 2 ℓ + 1 the same conclusion holds provided G is free from false twins. On the other hand, Spoiler always wins on $$G\circ P_5$$ G ∘ P 5 . In most of the cases, the corresponding Maker-Breaker resolving number is also determined.

A set of vertices S ⊆ V (G) is secure if it is possible to thwart an assault on any vertex set X ⊆ S. That is, a set S is secure if and only if it can fight off every attack. The security number of G, s(G), is the minimum cardinality of a secure set in G. In this paper, we initiated a study of security in the Cartesian product of graphs and obtained a characterization for s(G□H) = 2 and 3. We have also improved the existing upper bound for s(G□H) for some classes of graphs.

In this paper, we study the problem of expressing the secure Italian domination number of lexicographic product graphs [Formula: see text] in terms of a domination parameter of the graph [Formula: see text]. For this purpose, we use the approach of [Formula: see text]-domination introduced in [A. Cabrera Martínez, A. Estrada-Moreno and J. A. Rodríguez-Velázquez, from Italian domination in lexicographic product graphs to [Formula: see text]-domination in graphs, Ars Math. Contemp. 22 (2022) P1.04], where [Formula: see text] is a vector whose components are integers and whose values depend on the value of some domination parameters of the graph [Formula: see text].

This paper introduces Einstein fuzzy graphs as a novel framework for representing and analyzing uncertain or imprecise relationships between entities using fuzzy set theory. By employing the Einstein t-norm and t-conorm operations, fuzzy analogs of fundamental graph concepts are developed, emphasizing both their theoretical depth and practical relevance. A key contribution of this work is the introduction of modified product operations, which are carefully designed to ensure that the product of any two Einstein fuzzy graphs results in a valid Einstein fuzzy graph. These modified operations are shown to be more suitable than traditional product operations when dealing with the nuanced behavior of fuzzy relationships. Definitions for various types of graph products are presented, along with illustrative examples. The properties of these products are thoroughly analyzed and comparisons with traditional operations underscore the advantages of the modified approach. Also, key propositions and theorems are established to support a comprehensive understanding of Einstein fuzzy graphs and their structure. Real-life applications are also discussed, demonstrating the practical utility of this framework.

The lower bounds of 4-tree connectivity of Cartesian product graphs

Cells are organized to form three-dimensional structures of complex tissues. To map the complete 3D organization of a tissue, technologies based on tissue microdissections provide deep bulk RNA sequencing of orthogonally arranged cryosections of a tissue, such that the full 3D spatial structure could be inferred from deeply sequenced transcriptomes in three views projected similarly as 3D tomography. Here, we introduce CTFacTomo to learn a Collapsed Tensor Factorization for RNA tomography data from cryosections to reconstruct 3D spatially resolved gene expressions. CTFacTomo combines tensor factorization with collapsing tensor entries to match the bulk gene expressions in each cryosection, enriched by a regularization of a product graph of protein-protein interaction network and spatial graphs. In the experiments, CTFacTomo is first validated on three datasets projected from fully profiled 3D spatial gene expressions to demonstrate that CTFacTomo significantly outperforms the benchmark methods for predicting the ground-truth gene expressions based on the projected 1D spatial gene expressions of three orthographic views. CTFacTomo is then applied to two RNA tomography datasets from zebrafish embryo and mouse olfactory mucosa, respectively. In both datasets, CTFacTomo detects 3D spatial expressions of several marker genes that are consistent with the developmental or functional regions in comparison to accompanying ISH staining images. In addition, a qualitative comparison between the reconstructed zebrafish embryo gene expressions with a matched external 3D Stereo-seq dataset also suggests that CTFacTomo reconstructs more spatially coherent patterns in the whole transcriptome with state-of-the-art performance.

We show that every n-vertex planar graph is contained in the graph obtained from a fan by blowing up each vertex by a complete graph of order O(√(n log² n)). Equivalently, every n-vertex planar graph G has a set X of O(√(n log² n)) vertices such that G−X has bandwidth O(√(n log² n)). We in fact prove the same result for any proper minor-closed class, and we prove more general results that explore the trade-off between X and the bandwidth of G−X. The proofs use three key ingredients. The first is a new local sparsification lemma, which shows that every n-vertex planar graph G has a set of O((n log n)/δ) vertices whose removal results in a graph with local density at most δ. The second is a generalization of a method of Feige and Rao that relates bandwidth and local density using volume-preserving Euclidean embeddings. The third ingredient is graph products, which are a key tool in the extension to any proper minor-closed class.

We show that every n-vertex planar graph is contained in the graph obtained from a fan by blowing up each vertex by a complete graph of order O(√(n log² n)). Equivalently, every n-vertex planar graph G has a set X of O(√(n log² n)) vertices such that G−X has bandwidth O(√(n log² n)). We in fact prove the same result for any proper minor-closed class, and we prove more general results that explore the trade-off between X and the bandwidth of G−X. The proofs use three key ingredients. The first is a new local sparsification lemma, which shows that every n-vertex planar graph G has a set of O((n log n)/δ) vertices whose removal results in a graph with local density at most δ. The second is a generalization of a method of Feige and Rao that relates bandwidth and local density using volume-preserving Euclidean embeddings. The third ingredient is graph products, which are a key tool in the extension to any proper minor-closed class.

Metric and strong metric dimension in intersection graphs of subgroups of finite groups

Moffatt in his paper Excluded minors and the ribbon graphs of knots (Journal of Graph Theory, 2016), conjectures that every ribbon graph minor-closed family can be characterized by a finite set of excluded ribbon graph minors. He supports this conjecture in several papers, particularly in Ribbon graph minors and low-genus partial duals (Annals of Combinatorics, 2016), by giving a finite list of excluded minors that characterizes the class of ribbon graphs with a partial dual of Euler genus at most one. In this paper, we give a finite list of excluded minors that characterizes ribbon graphs with a partial dual of Euler genus at most two, subject to the condition that any bouquet related by partial duality to the ribbon graph satisfies that the intersection graph of the induced subgraph of its non-orientable loops and the complement of the intersection graph of the induced subgraph of its orientable loops are both $3$-cycle free.

On Intersection Graphs of Spherical Caps

This study examines the δ-complements of graphs—a specific type of graph complement whose adjacency depends on the adjacency of the vertices with identical degrees in the original graph. More specifically, we study this type of complement regarding the domination number. We provide sharp Nordhaus–Gaddum-type bounds on the domination number of a graph and its δ-complement. We also provide sharp bounds on the domination numbers of the δ-complements of joined graphs and Cartesian product graphs.

Analyzing the structure of the automorphism groups of graphs, and investigating the properties of graphs that are constructed from algebraic structures are two important research topics in algebraic graph theory. Blending these two aspects of study, an algebraic intersection graph, called the invariant intersection graph of a graph, has been introduced in the literature. In this paper, we study certain properties of the invariant intersection graphs of graphs, and obtain some structural characterizations of these graphs, based on the automorphism group of the graph on which the invariant intersection graph is constructed.