Polygonal chains with minimal energy

Linear Algebra and its Applications, 2009

The energy of a graph is defined as the sum of the absolute values of all the eigenvalues of the graph. Let G(n, d) be the class of tricyclic graphs G on n vertices with diameter d and containing no vertex disjoint odd cycles C p , C q of lengths p and q with p + q ≡ 2 (mod 4). In this paper, we characterize the graphs with minimal energy in G(n, d). Lemma 1.1 [25]. Let G be any graph. Then b 4 (G) = m(G, 2) − 2s, where m(G, 2) is the number of 2-matchings of G and s is the number of quadrangles in G. Lemma 1.2 [15]. If G ∈ G(n, d), then b 2i 0 for 0 i n 2 .

Computational and Applied Mathematics

An eigenvalue of the adjacency matrix of a graph is said to be main if the all-1 vector is not orthogonal to the associated eigenspace. In this work, we approach the main eigenvalues of some graphs. The graphs with exactly two main eigenvalues are considered and a relation between those main eigenvalues is presented. The particular case of harmonic graphs is analyzed and they are characterized in terms of their main eigenvalues without any restriction on its combinatorial structure. We give a necessary and sufficient condition for a graph G to have -1λ min as an eigenvalue of its complement, where λ min denotes the least eigenvalue of G. Also, we prove that among connected bipartite graphs, K r,r is the unique graph for which the index of the complement is equal to -1λ min . Finally, we characterize all paths and all double stars (trees with diameter three) for which the smallest eigenvalue is non-main. Main eigenvalues of paths and double stars are identified.

2017

Let G = (V,E) be a simple graph. The energy of G is the sum of absolute values of the eigenvalues of its adjacency matrix A(G). In this paper we consider the edge energy of G (or energy of line of G) which is defined as the absolute values of eigenvalues of edge adjacency matrix of G. We study the edge energy of specific graphs.

2020

Let G be a simple, connected graph on the vertex set V(G) and the edge set E(G). For the degree of the vertex denoted by , the maximum degree is denoted by and the minimum degree is denoted by . If and are adjacent, then it is represented by . The adjacency matrix is a symmetric square matrix that determines the corner pairs in a graph. Let denote the eigenvalues of adjacency matrix. The greatest eigenvalue is said to as the spectral radius of the graph G. The energy of graph G is defined as . The Laplacian matrix of a graph G is represented by where is the degree matrix. The degree matrix is the diagonal matrix formed by the degree of each point belonging to G. The Laplacian eigenvalues are real. The graph laplacian energy is described by = with edges and vertices.

Mathematical Problems in Engineering, 2021

Let G be a graph with vertex set V G = v 1 , … , v n , and let d i be the degree of v i . The Zagreb matrix of G is the square matrix of order n whose i , j -entry is equal to d i + d j if the vertices v i and v j are adjacent, and zero otherwise. The Zagreb energy ZE G of G is the sum of the absolute values of the eigenvalues of the Zagreb matrix. In this paper, we determine some classes of Zagreb hyperenergetic, Zagreb borderenergetic, and Zagreb equienergetic graphs.

Let G = (V, E) be a simple graph of order n with m edges. The energy of a graph G, denoted by E(G), is defined as the sum of the absolute values of all eigenvalues of G. The Laplacian energy of the graph G is defined as

Canadian Journal of Mathematics, 1979

In this paper we investigate the action of finite groups G on finite polygonal graphs. The notion of a polygonal graph was introduced in [17]: A polygonal graph is a pair (J^, $) consisting of a graph ffl which is regular, connected and has girth m for some m ^ 3, and a set <$ of m-gons of ffl such that every 2-claw of Jf is contained in an unique element of S\ (See Section 2 for the définitions of the terms used here.) If < § is the set of all m-gons of ffl, so that there is in J^ an unique m-gon on every one of its 2-claws, then we write ffl for (Jf, <o) and call ffl a strict polygonal graph. If we wish to emphasize the integer m, then we call (Jti?, <f) an m-gon-graph (respectively, a strict m-gon-graph). Examples of polygonal graphs not arising from regular solids are known mainly with girth m S 6 and with valency k S 5. Fewer examples with m > 6 or k > 5 are known, the most notable arising from J u Janko's first simple group (m = 5 and k = 11), which in fact can be characterized by this action on a polygonal graph [15]. These examples will be discussed in Section 3. In Section 2 we define the terms used in this paper and prove some basic lemmas about strict polygonal graphs and their automorphism groups. In Sections 4 and 5 we shall assume that (J^, #) is a polygonal graph of valency k ^ 3 on a set 12, with girth m, m odd, m ^ 5, and that G S Aut (J^f) is a group of automorphisms of ffl transitive on 12. We also suppose that for any 2-claw (x:y, z), x, y, z £ 12, every involution in G xyz fixes (pointwise) the m-gon in S on (x'.y, z), but no other m-gon on (x'.y, z). This latter hypothesis is automatically satisfied if ffl is a strict m-gon-graph, and in the case that G xyz has no involutions w r e interpret this hypothesis to mean that G xyz fixes the m-gon in S on (x'.y, z), and no other m-gon on (x'.y, z). We shall then prove the following two theorems. THEOREM 1. Let x £ 12. Suppose that for some prime p and integer n > 0, PSL(2,p n) g G X A(X) S PTL(2,p n) on p n + 1 points.

Mathematics and Statistics, 2024

Energy of the graph G is the sum of absolute values of eigenvalues of its adjacency matrix. Given a simple connected graph G, its first (second) Zagreb matrix is constructed by including the sum (product) of the degrees of each pair of adjacent vertices of G. Computation of sum of absolute eigen values of these matrices yields the corresponding Zagreb energies. In this paper, the first and second Zagreb energies of certain families of graphs have been computed and a criterion to discern the nature of graph G based on their energies is obtained. The paper focuses on the comparative analysis of first and second Zagreb energies in terms of regular graphs such as cycle graphs, bipartite and tripartite graphs. Our findings reveal that the second Zagreb energy is always greater than first Zagreb energy for all complete bipartite graphs of even order greater than or equal to 4. Also we have established that the same is the case for complete tripartite graphs too. Furthermore, we illustrate that the two Zagreb energies coincide exclusively for the complete bipartite graph with equal partite sets if and only if the graph is of order 2. Additionally, we provide a criterion leading to an infinite set of non-isomorphic Zagreb equi-energetic graphs for all r > 1 within partite graphs. The computations of two Zagreb energies for graph operations like t-splitting graph and t-shadow graph are also illustrated. The first and second Zagreb energies for some specific graphs along with bounds on Zagreb energies for wheel graphs are also discussed.

Match-communications in Mathematical and in Computer Chemistry, 2020

Let G be a simple undirected graph with n vertices and m edges. The energy of G, E(G) corresponds to the sum of its singular values. This work obtains lower bounds for E(G) where one of them generalizes a lower bound obtained by Mc Clelland in 1971 to the case of graphs with given nullity. An extension to the bipartite case is given and, in this case, it is shown that the lower bound 2 √ m is improved. The equality cases are characterized. Moreover, a simple lower bound that considers the number of edges and the diameter of G is derived. A simple lower bound, which improves the lower bound 2 √ n− 1, for the energy of trees with n vertices and diameter d is also obtained. 1 Notation and Preliminaries In this work we deal with an (n,m)-graph G which is an undirected simple graph with vertex set V (G) and edge set E (G) of cardinality n and m, respectively. As usual we denote the adjacency matrix of G by A = A(G). The eigenvalues of G are the eigenvalues of A (see e.g. [5, 6]). Its eig...

2016

An eigenvalue of the adjacency matrix of a graph is said to be main if the all-1 vector is not orthogonal to the associated eigenspace. In this work, we approach the main eigenvalues of some graphs. The graphs with exactly two main eigenvalues are considered and a relation between those main eigenvalues is presented. The particular case of harmonic graphs is analyzed and they are characterized in terms of their main eigenvalues without any restriction on its combinatorial structure. We give a necessary and sufficient condition for a graph G to have −1 − λ min as an eigenvalue of its complement, where λ min denotes the least eigenvalue of G. Also, we prove that among connected bipartite graphs, K r,r is the unique graph for which the index of the complement is equal to −1 − λ min. Finally, we characterize all paths and all double stars (trees with diameter three) for which the smallest eigenvalue is non-main. Main eigenvalues of paths and double stars are identified.