A class of reflexive cactuses with four cycles

Treelike Comparability Graphs

For a simple finite graph G, the generalized adjacency matrix is defined as Aα(G)=αD(G)+(1−α)A(G),α∈[0,1], where A(G) and D(G) are respectively the adjacency matrix and diagonal matrix of the vertex degrees. The Aα-spread of a graph G is defined as the difference between the largest eigenvalue and the smallest eigenvalue of the Aα(G). In this paper, we answer the question posed in (Lin, Z.; Miao, L.; Guo, S. Bounds on the Aα-spread of a graph. Electron. J. Linear Algebra 2020, 36, 214–227). Furthermore, we show that the path graph, Pn, has the smallest S(Aα) among all trees of order n. We establish a relationship between S(Aα) and S(A). We obtain several bounds for S(Aα).

On distance Laplacian spectral radius and chromatic number of graphs

The Q-index of a simple graph is the largest eigenvalue of its signless Laplacian, or Q-matrix. In our previous paper [1] we gave three lower and three upper bounds for the Q-index of nested split graphs, also known as threshold graphs. In this paper, we give another two upper bounds, which are expressed as solutions of cubic equations (in contrast to quadratics from [1]). Some computational results are also included.

In this paper we consider the problem of approximating a consensus network by a less complex network, by removing cycles from the original network graph. The consensus network consists of agents that exchange relative state information with their neighbors in the network. We assume the agents have single-integrator dynamics and the network graph is undirected. The network used to approximate the original system has the same nodes as the original graph, but its edge set is a strict subset of the original edge set. We obtain a priori upper bounds on the absolute approximation error, depending on the length of the removed cycles, the algebraic connectivity of a chosen spanning tree of the network graph, and the largest eigenvalue of the Laplacian matrix of that spanning tree.

Let G be a simple graph of order n. Prime labeling is a bijective function f:V(G)→{1,2,…,n} such that gcd⁡(f(u),f(v))=1 for every pair of adjacent vertices u,v in G. A graph G that satisfies the definition of prime labeling is called a prime graph. The Dutch windmill graph D_r^n is a graph obtained by taking n copies of cycle graph C_r with a vertex in common. The double quadrilateral graph DQ is a graph constructed from two copies of C_4 and identifying one edge from each of them. The graph obtained by taking n copies of DQ and identifying one vertex of degree 3 from each of them as a common central vertex is called the double quadrilateral Dutch windmill graph DQ_n, for n≥1. Furthermore, graphs D_r^n and DQ_n becomes the base graph to construct two new graph classes, namely graph P_2 [D_r^n] and flower double quadrilateral graph FDQ_n. Both graph classes, constructed from the Dutch windmill graph, also contain even cycles. From previous research, it is known that graphs P_2 [D_4^n] and flower double quadrilateral graph FDQ_n have odd harmonious labeling. However, the determination of prime labeling on both classes is still an open problem. In this paper, we show that two classes of graphs constructed from Dutch windmill graphs with even cycles, namely graphs P_2 [D_4^n] and flower double quadrilateral graphs FDQ_n for n≥1 have a prime labeling. The result of this research shows that these graphs are prime graphs.

Given a simple (weighted) graph, or a collection of graphs on a common vertex set, we seek an assignment of vectors to the vertices such that the dot products of these vectors approximate the weight/frequency of the edges. By transforming vertices into (low dimensional) vectors, one can bring geometric methods to bear in the analysis of the graph(s). We illustrate our approach on the Mathematicians Collaboration Graph [Grossman (1996) The Erdős number project, http://www.oakland.edu/enp ] and the times series of Interstate Alliance Graphs (Gibler and Sarkees in J Peace Res 41(2):211–222, 2004).

Let G be an undirected simple graph. Graph coloring is a special case of labeling, and G is said to admit a proper coloring if no two neighbouring vertices of it are given the identical color. The vertices of identical color constitute a color class. A graph is p-colorable if it has a p-coloring. The chromatic number of G, denoted by χ(G), is the minimum p such that G is p-colorable. A graph G is equitably p-colorable if it has a p-coloring and the absolute difference in size between any two distinct color classes is at most 1. The equitable chromatic number of G, denoted by χ=(G), is the minimum p such that G is equitably p-colorable. The equitable chromatic threshold of G, denoted by , is the minimum p ′ such that G is equitably p-colorable for all p ≥ p ′ . A windmill graph Wnm consists of m copies of the complete graph Kn, with every vertex connected to a common vertex. In this paper, we give exact values of χ=(G) and when G is a windmill graph, bistar windmill graph, cycle windmill graph, and complete windmill graph.

We consider the following two classes of simple graphs: open necklaces and closed necklaces, consisting of a finite number of cliques of fixed orders arranged in path-like pattern and cycle-like pattern, respectively. In these two classes we determine those graphs whose index (the largest eigenvalue of the adjacency matrix) is maximal.