Edge Degree Weight of Sequential Join of Graphs

Let the weight w of an edge e= uv={u,v} of a graph G be defined by w(e)=deg(u)+deg(v)-2 and the weight of G be defined by w(G)= ∑ e ∈ E(G)w(e), where E(G) is the edge set of G. In this paper the weights of joins, sequential joins, unions, intersections, and products (Cartesian and Tensor) of sets of graphs are obtained. This leads to a variety of open questions and new studies

Nullity of a graph with a cut-edge

The nullity of a graph is known to be an analytical tool to predict reactivity and conductivity of molecular π-systems. In this paper we consider the change in nullity when graphs with a cut-edge, and others derived from them, undergo geometrical operations. In particular, we consider the deletion of edges and vertices, the contraction of edges and the insertion of an edge at a coalescence vertex. We also derive three inequalities on the nullity of graphs along the same lines as the consequences of the Interlacing Theorem. These results shed light, in the tight-binding source and sink potential model, on the behaviour of molecular graphs which allow or bar conductivity in the cases when the connections are either distinct or ipso.peer-reviewe

Coalescing Fiedler and core vertices

The nullity of a graph G is the multiplicity of zero as an eigenvalue in the spectrum of its adjacency matrix. From the interlacing theorem, derived from Cauchy’s inequalities for matrices, a vertex of a graph can be a core vertex if, on deleting the vertex, the nullity decreases, or a Fiedler vertex, otherwise. We adopt a graph theoretical approach to determine conditions required for the identification of a pair of prescribed types of root vertices of two graphs to form a cut-vertex of unique type in the coalescence. Moreover, the nullity of subgraphs obtained by perturbations of the coalescence G is determined relative to the nullity of G. This has direct applications in spectral graph theory as well as in the construction of certain ipso-connected nano-molecular insulators.peer-reviewe

The conductivity of superimposed key-graphs with a common one-dimensional adjacency nullspace

Two connected labelled graphs H1 and H2 of nullity one, with identical one-vertex deleted subgraphs H1 − z1 and H2 − z2 and having a common eigenvector in the nullspace of their 0-1 adjacency matrix, can be overlaid to produce the superimposition Z. The graph Z is H1 + z2 and also H2 + z1 whereas Z + e is obtained from Z by adding the edge {z1, z2}. We show that the nullity of Z cannot take all the values allowed by interlacing. We propose to classify graphs with two chosen vertices according to the type of the vertices occurring by using a 3-type-code. Out of the 27 values it can take, only 9 are hypothetically possible for Z, 8 of which are known to exist. Moreover, the SSP molecular model predicts conduction or insulation at the Fermi level of energy for 11 possible types of devices consisting of a molecule and two prescribed connecting atoms over a small bias voltage. All 11 molecular device types are realizable for general molecules, but the structure of Z and of Z + e restricts the number to just 5.peer-reviewe

Coalescing Fiedler and core vertices

summary:The nullity of a graph $G$ is the multiplicity of zero as an eigenvalue in the spectrum of its adjacency matrix. From the interlacing theorem, derived from Cauchy's inequalities for matrices, a vertex of a graph can be a core vertex if, on deleting the vertex, the nullity decreases, or a Fiedler vertex, otherwise. We adopt a graph theoretical approach to determine conditions required for the identification of a pair of prescribed types of root vertices of two graphs to form a cut-vertex of unique type in the coalescence. Moreover, the nullity of subgraphs obtained by perturbations of the coalescence $G$ is determined relative to the nullity of $G$. This has direct applications in spectral graph theory as well as in the construction of certain ipso-connected nano-molecular insulators