let $g$ be a non-abelian finite group. in this paper, we prove that $gamma(g)$ is $k_4$-free if and only if $g cong a times p$, where $a$ is an abelian group, $p$ is a $2$-group and $g/z(g) cong mathbb{ z}_2 times mathbb{z}_2$. also, we show that $gamma(g)$ is $k_{1,3}$-free if and only if $g cong {mathbb{s}}_3,~d_8$ or $q_8$.

we define minimal cn-dominating graph $mathbf {mcn}(g)$, commonality minimal cn-dominating graph $mathbf {cmcn}(g)$ and vertex minimal cn-dominating graph $mathbf {m_{v}cn}(g)$, characterizations are given for graph $g$ for which the newly defined graphs are connected. further serval new results are developed relating to these graphs.

Random walk kernels in conjunction with Support Vector Machines are powerful methods for error-tolerant graph matching. Because of their local definition, however, the applicability of random walk kernels strongly depends on the characteristics of the underlying graph representation. In this paper, we describe a simple extension to the standard random walk kernel based on graph edit distance. T...

We discuss 11 known basic models of distributed computing: four message-passing models that differ by the (non)existence of port-numbers and a hierarchy of seven local computations models. In each of these models, we study the computational complexity of the decision problems if the leader election and if the naming problem can be solved on a given network. It is already known that these two de...

The cycle and co-cycle matroids of a bond graph are defined using chain group matroids derived,from the cycle and co-cycle vector spaces qf a bond graph. The relationship between these structures is investigated and various results are proved. A precise equivalence is dejined for bond graphs. Duality theory is seen to be very clear in the context of bond graph matroids.

Let G be a connected graph and r a group of automorphisms of G. We enumerate the number of r-isomorphism classes of derived graph coverings of G with voltages in a finite field of prime order P (>2).

Given a planar graph derived from a spherical, euclidean or hyperbolic tessellation, one can define a discrete curvature by combinatorial properties, which after embedding the graph in a compact 2d-manifold, becomes the Gaussian curvature .

We show that an important recent result of Alon and Shapira on testing hereditary graph properties can be derived from the existence of a limit object for convergent graph sequences.

In this paper, we enumerate the parameter matrices of all perfect $2$-colorings of the Platonic graphs consisting of the tetrahedral graph, the cubical graph, the octahedral graph, the dodecahedral graph, and  the icosahedral graph.

For ??[0,1], let A?(G?)=?D(G)+(1??)A(G?), where G is a simple undirected graph, D(G) the diagonal matrix of its vertex degrees and A(G?) adjacency signed graph G? whose underlying G. In this paper, basic properties A?(G?) are obtained, positive semidefiniteness studied some bounds on eigenvalues derived—in particular, lower upper largest eigenvalue obtained.