altan derivatives of polycyclic conjugated hydrocarbons were recently introduced and studied in theoretical organic chemistry. we now provide a generalization of the altan concept, applicable to any graph. several earlier noticed topological properties of altan derivatives of polycyclic conjugated hydrocarbons are shown to be the properties of all altan derivatives of all graphs. among these ar...

Let G be a graph with n vertices and ν(G) be the matching number of G. The inertia of a graph G, In(G) = (n+, n−, n0) is an integer triple specifying the numbers of positive, negative and zero eigenvalues of the adjacency matrix A(G), respectively. Let η(G) = n0 denote the nullity of G (the multiplicity of the eigenvalue zero of G). It is well known that if G is a tree, then η(G) = n− 2ν(G). Gu...

Let Gr,p be a graph obtained from a path by adjoining a cycle Cr of length r to one end and the central vertex of a star Sp on p vertices to the other end. In this paper, it is proven that unicyclic graph Gr,p with r even is determined by its Laplacian spectrum except for n = p+4.

We prove that the minimum value of the least eigenvalue of the signless Laplacian of a connected nonbipartite graph with a prescribed number of vertices is attained solely in the unicyclic graph obtained from a triangle by attaching a path at one of its endvertices. © 2008 Elsevier Inc. All rights reserved. AMS classification: 05C50

Altan derivatives of polycyclic conjugated hydrocarbons were recently introduced and studied in theoretical organic chemistry. We now provide a generalization of the altan concept, applicable to any graph. Several earlier noticed topological properties of altan derivatives of polycyclic conjugated hydrocarbons are shown to be the properties of all altan derivatives of all graphs. Among these ar...

Let Gr,p be a graph obtained from a path by adjoining a cycle Cr of length r to one end and the central vertex of a star Sp on p vertices to the other end. In this paper, it is proven that unicyclic graph Gr,p with r even is determined by its Laplacian spectrum except for n = p+4.

Let G be a simple connected graph with n vertices and n edges which we call an unicyclic graph. In this paper, we first investigate the least eigenvalue λn(G), then we present two sharp bounds of the spread s(G) of G. AMS classification: 05C50 ,05C35

the reverse degree distance of a connected graph $g$ is defined in discrete mathematical chemistry as [ r (g)=2(n-1)md-sum_{uin v(g)}d_g(u)d_g(u), ] where $n$, $m$ and $d$ are the number of vertices, the number of edges and the diameter of $g$, respectively, $d_g(u)$ is the degree of vertex $u$, $d_g(u)$ is the sum of distance between vertex $u$ and all other vertices of $g$, and $v(g)$ is the ...

The Balaban index of a connected graph G is defined as J(G) = |E(G)| μ+ 1 ∑ e=uv∈E(G) 1 √ DG(u)DG(v) , and the Sum-Balaban index is defined as SJ(G) = |E(G)| μ+ 1 ∑ e=uv∈E(G) 1 √ DG(u)+DG(v) , where DG(u) = ∑ w∈V (G) dG(u,w), and μ is the cyclomatic number of G. In this paper, the unicyclic graphs with the maximum Balaban index and the maximum Sum-Balaban index among all unicyclic graphs on n v...

Let G be a graph on n vertices and let λ1, λ2, . . . , λn be its eigenvalues. The energy of G is defined as E(G) = |λ1| + |λ2| + · · · + |λn|. For various classes of unicyclic graphs, the graphs with maximal energy are determined. Let P 6 n be obtained by connecting a vertex of the circuit C6 with a terminal vertex of the path Pn−6. For n 7, P 6 n has the maximal energy among all connected unic...