[Show abstract][Hide abstract] ABSTRACT: Keywords: Laplacian eigenvalues Laplacian energy Vertex connectivity Edge connectivity Vertex cover number Spanning tree packing number a b s t r a c t For G being a graph with n vertices and m edges, and with Laplacian eigenvalues μ 1 ≥ μ 2 ≥ · · · ≥ μ n−1 ≥ μ n = 0, the Laplacian energy is defined as LE = n i=1 |μ i − 2m/n|. Let σ be the largest positive integer such that μ σ ≥ 2m/n. We characterize the graphs satisfying σ = n − 1. Using this, we obtain lower bounds for LE in terms of n, m, and the first Zagreb index. In addition, we present some upper bounds for LE in terms of graph invariants such as n, m, maximum degree, vertex cover number, and spanning tree packing number.
[Show abstract][Hide abstract] ABSTRACT: In a study on the structure--dependency of the total $\pi$-electron energy
from 1972, Trinajsti\'c and one of the present authors have shown that it
depends on the sums $\sum_{v\in V}d(v)^2$ and $\sum_{v\in V}d(v)^3$, where
$d(v)$ is the degree of a vertex $v$ of the underling molecular graph $G$. The
first sum was later named {\it first Zagreb index} and over the years became
one of the most investigated graph--based molecular structure descriptors. On
the other hand, the second sum, except in very few works on the general first
Zagreb index and the zeroth--order general Randi\'c index, has been almost
completely neglected. Recently, this second sum was named {\it forgotten
index}, or shortly the \F$-{\it index}, and shown to have an exceptional
applicative potential. In this paper we examine the trees extremal with respect
to the $F$-index.
[Show abstract][Hide abstract] ABSTRACT: Applying the Cauchy–Schwarz inequality, we obtain a sharp upper bound on the Randić energy of a bipartite graph and of graphs whose adjacency matrix is partitioned into blocks with constant row sum.
Linear Algebra and its Applications 08/2015; 478. DOI:10.1016/j.laa.2015.03.039 · 0.94 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Graph invariants, based on the distances between the vertices of a graph, are widely used in theoretical chemistry. The degree resistance distance of a graph is defined as , where is the degree of the vertex , and the resistance distance between the vertices and . Let be the set of all cacti possessing vertices and cycles. The elements of with minimum degree resistance distance are characterized.
[Show abstract][Hide abstract] ABSTRACT: Given a graph , the atom–bond connectivity ( ) index is defined to be where and are vertices of , denotes the degree of the vertex , and indicates that and are adjacent. Although it is known that among trees of a given order , the star has maximum index, we show that if , then the star of order has minimum index among trees with leaves. If , then the balanced double star of order has the smallest index.
[Show abstract][Hide abstract] ABSTRACT: The resolvent Estrada index of a (non-complete) graph $G$ of order $n$ is defined as $EE_r =\sum_{i=1}^n(1-\lamda_i/(n-1))^{-1}$, where $\lamda_1, \lamda_2, \lamda_n$ are the eigenvalues of $G$. Combining computational and mathematical approaches, we establish a number of properties of $EE_r$. In particular, any tree has smaller $EE_r$-value than any unicyclic graph of the same order, and any unicyclic graph has smaller $EE_r$-value than any tricyclic graph of the same order. The trees, unicyclic, bicyclic, and tricyclic graphs with smallest and greatest $EE_r$ are determined.
MATCH Communications in Mathematical and in Computer Chemistry 05/2015; 74:431-440. · 1.47 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: In a recent paper [H. Lin,MATCHCommunications in Mathematical and in Computer Chemistry
70 (2013) 575–582], a congruence relation forWiener indices of a class of trees was reported. We now show
that Lin’s congruence is a special case of a much more general result.
[Show abstract][Hide abstract] ABSTRACT: A direct method for computation of the energy-effect (ef) of cycles in conjugated molecules is elaborated, based on numerical calculation of the (complex) zeros of certain graph polynomials. Accordingly, the usage of the Coulson integral formula can be avoided, and thus the ef-values can be calculated for arbitrary cycles of arbitrary conjugated systems.
[Show abstract][Hide abstract] ABSTRACT: 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.
[Show abstract][Hide abstract] ABSTRACT: Benzenoid molecules possessing bays are traditionally considered as “strain–free”. Yet, repulsion between the two bay H-atoms affects the length of the near-lying carbon–carbon bonds. A method is developed to estimate the energy of this strain. In the case of phenanthrene its value was found to be about 7 kJ/mol.
Chemical Physics Letters 02/2015; 625. DOI:10.1016/j.cplett.2015.02.039 · 1.90 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: The graphs and trees with smallest resolvent Estrada indices (EEr) are characterized. The connected graph of order n with smallest EEr-value is the n-vertex path. The second-smallest such graph is the (n-1)-vertex path with a pendent vertex attached at position 2. The tree with third-smallest EEr is the (n-1)-vertex path with a pendent vertex attached at position 3, conjectured to be also the connected graph with third-smallest EEr. Based on a computer-aided search, we established the structure of a few more trees with smallest EEr.
MATCH Communications in Mathematical and in Computer Chemistry 01/2015; 73(1):267-270. · 1.47 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: The energy ε(G) of a graph G is defined as the sum of the absolute values of the eigenvalues of its adjacency matrix. A graph G of order n is said to be borderenergetic if its energy equals the energy of the complete graph Kn , i.e., if ε(G) = 2(n - 1). We first show by examples that there exist connected borderenergetic graphs, different from the complete graph Kn . The smallest such graph is of order 7. We then show that for each integer n , n ≥ 7, there exists borderenergetic graphs of order n, different from Kn , and describe the construction of some of these graphs.
[Show abstract][Hide abstract] ABSTRACT: Inarecentpaper[H.Lin,MATCHCommunicationsinMathematicalandinComputerChemistry 70 (2013) 575–582], a congruence relation for Wiener indices of a class of trees was reported. We now show that Lin’s congruence is a special case of a much more general result.
[Show abstract][Hide abstract] ABSTRACT: For a simple connected graph G of order n, the Laplacian-energy-like invariant and the Kirchhoff index are calculated by LEL(G) = Sigma(n-1)(i=1) root mu(i); and K f(G) = n Sigma(n-1)(i=1) 1/mu(i), respectively, where mu(1,)mu 2,....,mu(n-1),mu(n)= 0 are the Laplacian eigenvalues of G. We obtain a sharp upper bound for K f and a sharp lower bound for LEL. Further, we obtain upper and lower bounds for LEL and K f for graphs C(G) (the clique-inserted graph or para-line graph), R(G) (obtained by changing each edge of G into a triangle), and H(G) (obtained by inserting a new vertex on each edge of G and by joining two new vertices if they lie on adjacent edges of G), as well as for the line graph of a semiregular graph.
MATCH Communications in Mathematical and in Computer Chemistry 01/2015; 73(1):41-59. · 1.47 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Let G be a simple graph with n vertices and m edges. Let d(i) be the degree of the i-th vertex of G. The Randic matrix R = (r(ij)) is defined by r(ij) = 1/root d(i)d(j) if the i-th and j-th vertices are adjacent and r(ij) = 0 otherwise. The Randic energy RE is the sum of absolute values of the eigenvalues of R. Cavers at al. [On the normalized Laplacian energy and general Randic index R-1(G) of graphs, Lin. Algebra Appl. 433 (2010) 172-190] obtained some bounds on RE, but did not characterize the extremal graphs. We now find these extremal graphs. Additional lower and upper bounds for RE are obtained, in terms of n, m, maximum degree Delta, minimum degree delta, and the determinant of the adjacency matrix.
MATCH Communications in Mathematical and in Computer Chemistry 01/2015; 73(1):81-92. · 1.47 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: The classical first and second Zagreb indices of a graph GG are defined as M1=∑vdv2 and M2=∑uvdudv, where dvdv is the degree of the vertex vv of GG. So far, the difference of M1M1 and M2M2 has not been studied. We show that this difference is closely related to the vertex-degree-based invariant RM2=∑uv(du−1)(dv−1)RM2=∑uv(du−1)(dv−1), and determine a few basic properties of RM2RM2.
[Show abstract][Hide abstract] ABSTRACT: The entropy of a graph is an information-theoretic quantity which expresses
the complexity of a graph \cite{DM1,M}. After Shannon introduced the definition
of entropy to information and communication, many generalizations of the
entropy measure have been proposed, such as R\'enyi entropy and Dar\`oczy's
entropy. In this article, we prove accurate connections (inequalities) between
generalized graph entropies, distinct graph energies and topological indices.
Additionally, we obtain some extremal properties of nine generalized graph
entropies by employing distinct graph energies and topological indices.