In 1972, within a study of the structure-dependency of total $$\pi $$
 -electron energy (
 $${\mathcal {E}}$$
 ), it was shown that $${\mathcal {E}}$$
 depends on the sum of squares of the vertex degrees of the molecular graph (later named first Zagreb index), and thus provides a measure of the branching of the carbon-atom skeleton. In the same paper, also the sum of cubes of degrees of vertices of the molecular graph was shown to influence $${\mathcal {E}}$$
 , but this topological index was never again investigated and was left to oblivion. We now establish a few basic properties of this “forgotten topological index” and show that it can significantly enhance the physico-chemical applicability of the first Zagreb index.

A forgotten topological index

In 1985, Fink and Jacobson gave a generalization of the concepts of domination and independence in graphs For a positive integer k, a subset S of vertices in a graph G = (V, E) is k-dominating if every vertex of V − S is adjacent to at least k vertices in S The subset S is k-independent if the maximum degree of the subgraph induced by the vertices of S is less or equal to k − 1 In this paper we survey results on k-domination and k-independence

k -Domination and k -Independence in Graphs: A Survey

We consider only simple graphs. Given two graphs $G$ with vertices $1,\ldots,n$ and $H$, the corona $G\circ H$ is defined as the graph obtained by taking $n$ copies of $H$ and for each $i$ inserting edges between the $i$th vertex of $G$ and each vertex of the $i$th copy of $H$. For a connected graph $G$ and any $r$-regular graph $H$ we provide complete information about the spectrum of $G\circ H$ using the spectrum of $G$ and spectrum of $H$. Complete information about the Laplacian spectrum of $G\circ H$ is also provided even when $H$ is not regular. A graph $G$ is said to have the property (R) if $\frac{1}{\lambda}$ is an eigenvalue of $G$ whenever $\lambda$ is an eigenvalue of $G$. Further, if $\lambda$ and $\frac{1}{\lambda}$ have the same multiplicity, for each eigenvalue $\lambda$, then it is said to have the property (SR). We characterize all trees with property (SR) and show that such a tree is the corona product of some tree and an isolated vertex. We supply a family of bipartite graphs with property (R). As an application we construct infinitely many pairs of nonisomorphic graphs with the same spectrum and the same Laplacian spectrum. We prove some results about the eigenvector related to the second smallest eigenvalue of the Laplacian matrix of $G\circ H$ and give an application.

The Spectrum of the Corona of Two Graphs

Theory and Applications of Graphs

On vertex symmetric digraphs

Our object in this note is to construct a new and simple operation on two graphs Gt and G2, called their corona, with the property that the group of the new graph is in general isomorphic with the wreath product of the groups of Ga and of G2. Consider two permutat ion groups A and B of order m and n respectively acting on objects sets X = { x 1, x2 ..... Xd} and Y={Yl,Yz,. . . ,Y~}. By their composition or wreath product A [B] we will mean that permutat ion group of order mn a acting on X × Y in which each permutat ion c~A and each sequence (/3~,/~2 . . . . . /~d) of permutations in B induce the permutat ion y =(~;/~1, f12 . . . . . /3a) such that Y(xi, y j) =(c~xi,/~iyj). We write A = B to mean that two permutat ion groups A and B are not only isomorphic but also permutat ionally equivalent. More specifically let h : A ~ B be an isomorphism. To define A = B, we also require another 1 1 map f : X~--~Y between the objects such that for all x in X and c~ in A, f (~x )=h(~) f (x ) . Let graphs 2) G 1 and G 2 have point sets 1/1 and V 2. Then GI[G2], their lexicographic product, has V1 x V2 as its set of points, with two of its points u = (u~, uz) and v =(v l , vz) adjacent whenever ul is adjacent to v~ in Gi, or Ux = v l and u2 is adjacent to v2 in G2. Let A~ and Az be the groups of graphs G~ and G2; then necessary and sufficient condit ions for A ~ CA/] to be permutationally equivalent with the group of GI [G2] were found by Sabidussi [2]. In order to state this result, we recall from [1] that F(G) denotes the group of graph G and (7 the complement of G, that the neighborhood N(v) of a point v of G is the set of points adjacent with v, and the closed neighborhood of v is N(v) u {v}.

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On the Corona of Two Graphs.

1. Introduction. For integers n and k with 2 ≤ 2k < n, the generalized Petersen graph G(n, k) has been defined in (8) to have vertex-setand edge-set E(G(n, k)) to consist of all edges of the formwhere i is an integer. All subscripts in this paper are to be read modulo n, where the particular value of n will be clear from the context. Thus G(n, k) is always a trivalent graph of order 2n, and G(5, 2) is the well known Petersen graph. (The subclass of these graphs with n and k relatively prime was first considered by Coxeter ((2), p. 417ff.).)

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0305004100049811

The groups of the generalized Petersen graphs

Abstract A permutation group A of degree n acting on a set X has a certain number of orbits, each a subset of X. More generally, A also induces an equivalence relation on X(k) the set of all k subsets of X, and the resulting equivalence classes are called k orbits of A, or generalized orbits. A self-complementary k-orbit is one in which for every k-subset S in it, X—S is also in it. Our main results are two formulas for the number s(A) of self-complementary generalized orbits of an arbitrary permutation group A in terms of its cycle index. We show that self-complementary graphs, digraphs, and relations provide special classes of self-complementary generalized orbits.

Self-complementary generalized orbits of a permutation group

Upper bounds (and in some cases the exact values) are given for the least number of lines a graph must have if its group of automorphisms is to be isomorphic to the cyclic group of order n.

Roberto Frucht

Papers

On the Corona of Two Graphs.

The groups of the generalized Petersen graphs

Self-complementary generalized orbits of a permutation group

Line-minimal Graphs with Cyclic Group