We focus on a specific class of planar cubic bridgeless graphs, namely the klee-graphs. K4 is the smallest klee-graph and having a klee-graph G we create another klee-graph by replacing any vertex of G by a triangle, i.e., applying Y∆ operation. In this paper, we prove that every klee-graph with n ≥ 8 vertices has at least 3 · 2 perfect matchings, improving the 2 bound inherited from the genera...

In 1949 R. Frucht established that any finite group can be represented by a graph of degree three in the sense that automorphism group of the graph is isomorphic to the given group. In 1957 G. Sabidussi gave another step in this direction, proving that every finite group can be represented by an infinite number of regular graphs. The main result of this work is to give a proof of the Sabidussi’...

Data structures are presented for the problem of maintaining a. minimum spanning tree on-line under the operation of updating the cost of some edge in the graph. For the case of a general graph, maintaining the data structure and updating the tree are shown to take O(vm) time. where m is the number of edges in the graph. For the case of a planar graph, a data structure is presented which suppor...

Similarity of graphs with labeled vertices and edges is naturally defined in terms of maximal common subgraphs. To avoid computation overload, a parameterized technique for approximation of graphs and their similarity is used. A lattice-based method of binarizing labeled graphs that respects the similarity operation on graph sets is proposed. This method allows one to compute graph similarity b...

Godsil-McKay switching is an operation on graphs that doesn’t change the spectrum of the adjacency matrix. Usually (but not always) the obtained graph is non-isomorphic with the original graph. We present a straightforward sufficient condition for being isomorphic after switching, and give examples which show that this condition is not necessary. For some graph products we obtain sufficient con...

Graph algebras establish a connection between directed graphs without multiple edges and special universal algebras of type (2, 0). We say that a graph G satisfies an identity s ≈ t if the corresponding graph algebra A(G) satisfies s ≈ t. A graph G = (V, E) is called an (xy)x ≈ x(yy) graph if the graph algebra A(G) satisfies the equation (xy)x ≈ x(yy). An identity s ≈ t of terms s and t of any ...

Graph algebras establish a connection between directed graphs without multiple edges and special universal algebras of type (2,0). We say that a graph G satisfies an identity s ≈ t if the corresponding graph algebra A(G) satisfies s ≈ t. A graph G is called associative if the corresponding graph algebra A(G) satisfies the equation (xy)z ≈ x(yz). An identity s ≈ t of terms s and t of any type τ ...

The recursive computation of the interlace polynomial introduced by Arratia, Bollobás and Sorkin is defined in terms of a new pivoting operation on undirected simple graphs. In this paper, we interpret the new pivoting operation on graphs in terms of standard pivoting (on matrices). Specifically, we show that, up to swapping vertex labels, Arratia et al.’s pivoting operation on a graph is equiv...

Probe machine (PM) is a recently reported mathematic model with massive parallelism. Herein, we presented searching the maximum clique of an undirected graph six vertices. We constructed data library containing n sublibraries, each sublibrary corresponded to vertex in given graph. Then, probe according induced subgraph was designed order search and generate all maximal cliques. Subsequently, pe...

This paper presents a new approach for intuitive and effective design of periodic symmetric tiles. We observe that planar graphs can effectively represent symmetric tiles and graph drawing provides an intuitive paradigm for designing symmetric tiles. Moreover, based on our theoretical work to represent hexagonal symmetry by rectangular symmetry, we are able to present all symmetric tiles as gra...