On the Eccentricity Energy of Various Graph Products

Pseudo-one-factorizations of regular graphs of odd order — II. products of graphs

This work presents the employment of graph theory for cryptographic applications. A Cartesian product graph (CPG) is used to propose new symmetric encryption schemes. On proposed encryption schemes, the ciphertexts of the plaintexts are sent to the receiver entity as the CPGs. The CPG is created based on two graphs G and H that are generated depending on the random choices of the plaintext bits. These bits are used as vertices of graphs G and H. The purpose of using the CPG is to increase the security level of the symmetric encryption schemes in comparison with other encryption schemes. Two study cases of the proposed symmetric Cartesian product graphic encryption (SCPGE) schemes are presented as new experimental results. The security issues of the proposed SCPGE schemes are determined. The SCPGE schemes consider as new insights for more secure communications.

This paper is concerned with the linkedness of Cartesian products of complete graphs. A graph with at least 2 k vertices is k -linked if, for every set of 2 k distinct vertices organised in arbitrary k pairs of vertices, there are k vertex-disjoint paths joining the vertices in the pairs. We show that the Cartesian product K d 1 +1 × K d 2 +1 of complete graphs K d 1 +1 and K d 2 +1 is ⌊( d 1 + d 2 )/2⌋ -linked for d 1 , d 2 ≥ 2 , and this is best possible. This result is connected to graphs of simple polytopes. The Cartesian product K d 1 +1 × K d 2 +1 is the graph of the Cartesian product T ( d 1 ) × T ( d 2 ) of a d 1 -dimensional simplex T ( d 1 ) and a d 2 -dimensional simplex T ( d 2 ) . And the polytope T ( d 1 ) × T ( d 2 ) is a simple polytope , a ( d 1 + d 2 ) -dimensional polytope in which every vertex is incident to exactly d 1 + d 2 edges. While not every d -polytope is ⌊ d /2⌋ -linked, it may be conjectured that every simple d -polytope is. Our result implies the veracity of the revised conjecture for Cartesian products of two simplices.

We study the question of polytopality of graphs: when is a given graph the graph of a polytope? We first review the known necessary conditions for a graph to be polytopal, and we present three families of graphs which satisfy all these conditions, but which nonetheless are not graphs of polytopes.Our main contribution concerns the polytopality of Cartesian products of non-polytopal graphs. On the one hand, we show that products of simple polytopes are the only simple polytopes whose graph is a product. On the other hand, we provide a general method to construct (non-simple) polytopal products whose factors are not polytopal.

Fiber-complemented graphs — I: structure and invariant subgraphs

A graph G is k-factor-critical if G ? S has a perfect matching for any k-subset S of V(G). In this paper, we investigate the factor-criticality in Cartesian products of graphs and show that Cartesian product of an m-factor-critical graph and an n-factor-critical graph is $${(m+n+\varepsilon )}$$ -factor-critical, where $${\varepsilon = 0}$$ if both of m and n are even; $${\varepsilon = 1}$$ , otherwise. Moreover, this result is best possible.

This paper considers a certain fairly large class of graph invariants and shows that for any invariant in this class, there is a pair of nonisomorphic graphs that have the same invariant. Some explicit examples of such invariants are discussed.

Let $$G_1$$ and $$G_2$$ be two simple connected graphs. The properties of $$\alpha -$$eigenvalues of four coronal graphs of $$G_1$$ and $$G_2$$: $$G_1 \circ G_2, G_1\lozenge G_2$$, $$G_1 \odot G_2$$ and $$G_1 \circleddash G_2$$ have been studied. As an application, we construct infinitely many pairs of non-isomorphic graphs with the same $$\alpha $$-eigenvalues.

A very natural question raised about products of graphs is the following. Do there exist necessary and sufficient conditions on a pair (G,H) of graphs for their product to have some specified property? In particular, for which graphs is the product one-factorizable? Sufficient conditions have been investigated for the cartesian, lexicographic, and tensor products in [3], [4], and [5]. However, the conditions for the tensor product to be one-factorizable are significantly more scanty than those for the other products. The purpose of this paper is to correct this situation somewhat.

We give two combinatorial proofs of a product formula for the number of spanning trees of the $n$-dimensional hypercube. The first proof is based on the assertion that if one chooses a uniformly random rooted spanning tree of the hypercube and orient each edge from parent to child, then the parallel edges of the hypercube get orientations which are independent of one another. This independence property actually holds in a more general context and has intriguing consequences. The second proof uses some "killing involutions'' in order to identify the factors in the product formula. It leads to an enumerative formula for the spanning trees of the $n$-dimensional hypercube augmented with diagonals edges, counted according to the number of edges of each type. We also discuss more general formulas, obtained using a matrix-tree approach, for the number of spanning trees of the Cartesian product of complete graphs.

On the geodetic number and related metric sets in Cartesian product graphs

We present theoretical results when applying the Cartesian product of two Kuramoto models on different network topologies. By a detailed mathematical analysis, we prove that the dynamics on the Cartesian product graph can be described by the canonical equations as the Kuramoto model. We show that the order parameter of the Cartesian product is the product of the order parameters of the factors. On the product graph, we observe either continuous or discontinuous synchronization transitions. In addition, under certain conditions, the transition from an initially incoherent state to a coherent one is discontinuous, while the transition from a coherent state to an incoherent one is continuous, presenting a mixture state of first and second order synchronization transitions. Our numerical results are in a good agreement with the theoretical predictions. These results provide new insight for network design and synchronization control.

On the complexity of recognizing S-composite and S-prime graphs

In this paper we continue the investigation of total domination in Cartesian products of graphs first studied in (Henning, M.A., Rall, D.F. in Graphs Comb. 21:63–69, 2005). A set S of vertices in a graph G is a total dominating set of G if every vertex in G is adjacent to some vertex in S. The maximum cardinality of a minimal total dominating set of G is the upper total domination number of G, denoted by Γt(G). We prove that the product of the upper total domination numbers of any graphs G and H without isolated vertices is at most twice the upper total domination number of their Cartesian product; that is, Γt(G)Γt(H)≤2Γt(G □ H).

The study of perfectness, via the strong perfect graph conjecture, has given rise to numerous investigations concerning the structure of many particular classes of perfect graphs. In “Perfect Product Graphs” (Discrete Mathematics, Vol. 20, 1977, pp. 177‐‐186), G. Ravindra and K. R. Parthasarathy tried, but unfortunately without success, to characterize the perfectness of the cartesian product of graphs (see also MR No. 58‐‐10567, 1979). In this paper we completely characterize the graphs that are both nontrivial cartesian products and s‐strongly perfect.