For an integer k 2 2, a proper k-restraint on a graph G is a function from the vertex set of G to the set of k-colors. A graph G is amenably k-colorable if, for each nonconstant proper k-restraint r on G, there is a k-coloring c of G with c(v) # r(v) for each vertex v of G. A graph G is amenable if it is amenably k-colorable and k is the chromatic number of G. For any k Z= 3, there are infinite...

Graph structure is widely used to describe problems in different fields. Problems many areas, such as security and transportation, are among them. The can be solved using approaches similar the graph structure. independent set problem, which NP-Complete one of main theory modeling problems. implementation Independent problem with most significant possible number nodes called Maximum set. A lot ...

In this paper, we propose a randomization scheme, LORA (Link Obfuscation by Randomization), to obfuscate edge existence in graphs. Specifically, we extract the source graph’s hierarchical random graph model and reconstruct the released graph randomly with this model. We show that the released graph can preserve critical graph statistical properties even after a large number of edges have been r...

A graph G is called k-critical if G is k-chromatic but every proper subgraph of G has chromatic number at most k 1. In this paper the following result is proved. If G is a k-critical graph (k>~4) on n vertices, then 21E(G)I>(k 1)n ÷ ((k 3)/(k 2 3))n + k 4 where n>~k + 2 and n ~ 2 k 1. This improves earlier bounds established by Dirac (1957) and Gallai (1963). (~) 1998 Elsevier Science B.V. All ...

A graph G is called k-critical if x(G) = k and x(G e) -C x(G) for each edge e of G, where x denotes the chromatic number. T. Gallai conjectured that every k-critical graph of order n contains at most n complete (k l)-subgraphs. In 1987, Stiebitz proved Gallai’s conjecture in the case k = 4, and in 1992 Abbott and Zhou proved Gallai’s conjecture for all k > 5. In their paper, Abbott and Zhou ask...

The energy of a graph is the sum of the absolute values of the eigenvalues of its adjacency matrix. Two graphs are equienergetic if they have the same energy. We construct infinite families of graphs equienergetic with edge-deleted subgraphs.

This study presents some mathematical methods for estimating the critical properties of 40 different types of alkanes and their derivatives including critical temperature, critical pressure and critical volume. This algorithm used QSPR modeling based on graph theory, several structural indices, and geometric descriptors of chemical compounds. Multiple linear regression was used to estimate the ...

A spectral inverse problem concerns the reconstruction of parameters a parent graph from prescribed data subgraphs. Also referred to as P–NP Isomorphism Problem, Reconstruction or Exact Graph Matching, aim is seek sets determine uniquely. Other related problems, including Polynomial Problem (PRP), involve recovery invariants. The PRP seeks extract spectrum deck cards each showing vertex-deleted...

A graph G = (V, E) is called factor-critical if G = ∅ and G − v has a perfect matching for every vertex v ∈ V (G). A factor-critical graph G is tight (anti-tight, respectively) if for any v ∈ V (G), any perfect matching M in G − v, and any e ∈ M , |N (v) ∩ V (e)| = 1 (|N (v) ∩ V (e)| = 2, respectively), where N (v) denotes the neighborhood of v and V (e) denotes the set of vertices incident wit...