The thickness of the Kronecker product of graphs

The tensor product is a fundamental mathematical concept with applications spanning linear algebra, graph theory, quantum computing, and representation theory. In graph theory, the tensor product provides a framework for analyzing structural relationships, particularly through the study of complete graphs, which yield complex networks from simple structures. Closely related is the Kronecker product of matrices, an essential tool for investigating tensor products via adjacency matrices. The Kronecker product preserves key algebraic properties, including linearity, distributivity, and associativity, and has played a central role in matrix analysis, systems theory, and signal processing. This work presents the definitions and core properties of the tensor product, supported by illustrative examples with complete graphs, and explores the Kronecker product along with its fundamental properties and theorems. By combining theoretical foundations with applications, the study offers both conceptual insights and practical perspectives on these algebraic constructions.

The eccentricity matrix 𝔼(G) of a graph G is constructed from the distance matrix D(G) by only retaining the leading values in each row and each column and setting the rest of the elements in the corresponding row and column to zero. The article aims to study the concepts of energy and the inertia of the eccentricity matrices of graphs obtained by means of various graph products, like the strong product, the Cartesian product, and the Kronecker product, and well-established relations are derived in terms of their base graphs. Furthermore, we construct infinitely many pairs of non-isomorphic graphs having the same set of eigenvalues with respect to the eccentricity matrix.

Vertex vulnerability parameters of Kronecker products of complete graphs

Heterogeneous sensor-based condition monitoring in directed energy deposition

In this paper four different versions of the DFT (DFT-j,j=I,II,III,IV) are first introduced, Then, the relationship among four different versions of the DFT and their inherent properties are explored. The new real-multiplier FFT-j algorithms are proposed for all four versions of the N=2/sup m/ DFT. The algorithm formulae are derived, represented by Kronecker product and direct sum. Finally, the signal flowgraph for the length-2/sup 3/ FFT-j is given to illustrate the proposed algorithms. The computational complexity is analysed and a comparison is made with other existing real-multiplier FFT algorithms. The proposed algorithms require the minimum number of arithmetic operations and use real multipliers and allow in-place computation. Besides, the proposed algorithms have very simple and regular structure. They have been implemented by software and have been finding practical applications. >

The crossing number of a graph [Formula: see text], denoted by [Formula: see text], is defined to be the minimum number of crossings that arise among all its drawings in the plane. This concept has been of interest to many researchers who have studied it for many families of graphs. In this paper, we introduce the crossing number of Cartesian product of sunlet graph [Formula: see text] with path [Formula: see text]. Further, we prove that the crossing number of [Formula: see text] is equal to [Formula: see text], along with giving a conjecture for the general case. In addition, we utilize the vertex’s rotation concept in order to prove some necessary conditions for the complete bipartite graph [Formula: see text] to be optimal when it is drawn in the plane, by presenting an upper bound for the crossing number of any subgraph in it together with determining the exact number of crossings in case the vertices of subgraph have the same rotation.

Let G and H be connected graphs. The tensor product G + H is a graph with vertex set V(G+H) = V (G) X V(H) and edge set E(G + H) ={(a , b)(x , y)| ax ∈ E(G) & by ∈ E(H)}. The graph H is called the strongly triangular if for every vertex u and v there exists a vertex w adjacent to both of them. In this article the tensor product of G + H under some distancebased topological indices are investigated, when H is a strongly triangular graph. As a special case most of results given by Hoji, Luob and Vumara in [Wiener and vertex PI indices of Kronecker products of graphs, Discrete Appl. Math., 158 (2010), 1848-1855] will be deduced.

In this paper we characterize the differentiating-dominating sets in the join, corona, and lexicographic product of graphs. We also determine bounds or the exact differentiating-domination numbers of these graphs.

A longest sequence $S$ of distinct vertices of a graph $G$ such that each vertex of $S$ dominates some vertex that is not dominated by its preceding vertices, is called a Grundy dominating sequence; the length of $S$ is the Grundy domination number of $G$. In this paper we study the Grundy domination number in the four standard graph products: the Cartesian, the lexicographic, the direct, and the strong product. For each of the products we present a lower bound for the Grundy domination number which turns out to be exact for the lexicographic product and is conjectured to be exact for the strong product. In most of the cases exact Grundy domination numbers are determined for products of paths and/or cycles.

Tree exploration with advice

An ordinary circle of a set P of n points in the plane is defined as a circle that contains exactly three points of P. We show that if P is not contained in a line or a circle, then P spans at least n^2/4 - O(n) ordinary circles. Moreover, we determine the exact minimum number of ordinary circles for all sufficiently large n and describe all point sets that come close to this minimum. We also consider the circle variant of the orchard problem. We prove that P spans at most n^3/24 - O(n^2) circles passing through exactly four points of P. Here we determine the exact maximum and the extremal configurations for all sufficiently large n. These results are based on the following structure theorem. If n is sufficiently large depending on K, and P is a set of n points spanning at most Kn^2 ordinary circles, then all but O(K) points of P lie on an algebraic curve of degree at most four. Our proofs rely on a recent result of Green and Tao on ordinary lines, combined with circular inversion and some classical results regarding algebraic curves.

For a simple connected graph G, let c : V(G) → ℕ be a vertex coloring of G, where adjacent vertices may be colored the same. The neighborhood color set of a vertex v, denoted by NC(v), is the set of colors of the neighbors of v. The coloring c is called a set coloring provided that NC(u) ≠ NC(v) for every pair of adjacent vertices u and v of G. The minimum number of colors needed for a set coloring of G is referred to as the set chromatic number of G and is denoted by χs(G). In this work, the set chromatic number of graphs is studied in relation to the graph operation called middle graph. Our results include the exact set chromatic numbers of the middle graph of cycles, paths, star graphs, double-star graphs, and some trees of height 2. Moreover, we establish the sharpness of some bounds on the set chromatic number of general graphs obtained using this operation. Finally, we develop an algorithm for constructing an optimal set coloring of the middle graph of trees of height 2 under some assumptions.

On the complexity of rainbow coloring problems

The skewness of a graph is the minimum number of edges that have to be removed to leave a planar subgraph. This is complementary, and computationally equivalent, to the Maximum Planar Subgraph problem. In this paper we look at the problem of computing the skewness of a graph with a small cutset. We show how to express the skewness of a graph with a cutset of size at most 4 in terms of the skewnesses of several derived graphs obtained by cutting along that cutset and `stitching up' afterwards. We conclude with a discussion on possible applications to planarisation.

On heuristics for determining the thickness of a graph