scholarly journals On bounds for harmonic topological index

Filomat ◽  
2018 ◽  
Vol 32 (1) ◽  
pp. 311-317 ◽  
Author(s):  
Marjan Matejic ◽  
Igor Milovanovic ◽  
Emina Milovanovic
Keyword(s):  
Topological Index ◽  
Upper Bounds ◽  
Simple Graph ◽  
Graph Invariant ◽  
Lower And Upper Bounds ◽  
Harmonic Index

Let G=(V,E), V = {1,2,..., n}, E = {e1,e2,..., em}, be a simple graph with n vertices and m edges. Denote by d1 ? d2 ?... ? dn > 0 and d(e1) ? d(e2) ?... ? d(em), sequences of vertex and edge degrees, respectively. If i-th and j-th vertices of the graph G are adjacent, it is denoted as i ~ j. Graph invariant referred to as harmonic index is defined as H(G)= ? i~j 2/di+dj. Lower and upper bounds for invariant H(G) are obtained.

scholarly journals ESTRADA INDEX OF GENERAL WEIGHTED GRAPHS

2012 ◽  
Vol 88 (1) ◽  
pp. 106-112 ◽  
Author(s):  
YILUN SHANG
Keyword(s):  
Weighted Graph ◽  
Upper Bounds ◽  
Weighted Graphs ◽  
Graph Invariant ◽  
Lower And Upper Bounds ◽  
Spectral Moments ◽  
Estrada Index ◽  
Low Order

AbstractLet $G$ be a general weighted graph (with possible self-loops) on $n$ vertices and $\lambda _1,\lambda _2,\ldots ,\lambda _n$ be its eigenvalues. The Estrada index of $G$ is a graph invariant defined as $EE=\sum _{i=1}^ne^{\lambda _i}$. We present a generic expression for $EE$ based on weights of short closed walks in $G$. We establish lower and upper bounds for $EE$in terms of low-order spectral moments involving the weights of closed walks. A concrete example of calculation is provided.

scholarly journals Extremal Values of Randić Index among Some Classes of Graphs

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Ali Ghalavand ◽  
Ali Reza Ashrafi ◽  
Marzieh Pourbabaee
Keyword(s):  
Upper Bounds ◽  
Simple Graph ◽  
Randić Index ◽  
Lower And Upper Bounds ◽  
Randic Index ◽  
Chemical Graphs ◽  
Cyclic Graphs ◽  
Vertex Degrees ◽  
Extremal Values

Suppose G is a simple graph with edge set E G . The Randić index R G is defined as R G = ∑ u v ∈ E G 1 / deg G u deg G v , where deg G u and deg G v denote the vertex degrees of u and v in G , respectively. In this paper, the first and second maximum of Randić index among all n − vertex c − cyclic graphs was computed. As a consequence, it is proved that the Randić index attains its maximum and second maximum on two classes of chemical graphs. Finally, we will present new lower and upper bounds for the Randić index of connected chemical graphs.

A short note on inverse sum indeg index of graphs

2019 ◽  
pp. 2050152
Author(s):  
Selvaraj Balachandran ◽  
Suresh Elumalai ◽  
Toufik Mansour
Keyword(s):  
Short Note ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Zagreb Index ◽  
Connected Graphs ◽  
Harmonic Index ◽  
Second Zagreb Index

The inverse sum indeg index of a graph [Formula: see text] is defined as [Formula: see text], where [Formula: see text] is the degree of the vertex [Formula: see text]. In a recent paper, Pattabiraman [Inverse sum indeg index of graphs, AKCE Int. J. Graphs Combinat. 15(2) (2018) 155–167] gave some lower and upper bounds on [Formula: see text] index of all connected graphs in terms of Harmonic index, second Zagreb index and hyper Zagreb index. But some results were erroneous. In this note, we have corrected these results.

scholarly journals On inverse degree and topological indices of graphs

Filomat ◽  
2016 ◽  
Vol 30 (8) ◽  
pp. 2111-2120 ◽  
Author(s):  
Kinkar Das ◽  
Kexiang Xu ◽  
Jinlan Wang
Keyword(s):  
Maximum Degree ◽  
Minimum Degree ◽  
Upper Bounds ◽  
Topological Indices ◽  
Simple Graph ◽  
Lower And Upper Bounds ◽  
Inverse Degree ◽  
Abc Index

Let G=(V,E) be a simple graph of order n and size m with maximum degree ? and minimum degree ?. The inverse degree of a graph G with no isolated vertices is defined as ID(G) = ?n,i=1 1/di, where di is the degree of the vertex vi?V(G). In this paper, we obtain several lower and upper bounds on ID(G) of graph G and characterize graphs for which these bounds are best possible. Moreover, we compare inverse degree ID(G) with topological indices (GA1-index, ABC-index, Kf-index) of graphs.

scholarly journals Investigation of General Power Sum-Connectivity Index for Some Classes of Extremal Graphs

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Rui Cheng ◽  
Gohar Ali ◽  
Gul Rahmat ◽  
Muhammad Yasin Khan ◽  
Andrea Semanicova-Fenovcikova ◽  
...  
Keyword(s):  
Connected Graph ◽  
Topological Index ◽  
Upper Bounds ◽  
Connectivity Index ◽  
Extremal Graphs ◽  
Graph Invariant ◽  
Power Sum ◽  
General Power

In this work, we introduce a new topological index called a general power sum-connectivity index and we discuss this graph invariant for some classes of extremal graphs. This index is defined by Y α G = ∑ u v ∈ E G d u d u + d v d v α , where d u and d v represent the degree of vertices u and v , respectively, and α ≥ 1 . A connected graph G is called a k -generalized quasi-tree if there exists a subset V k ⊂ V G of cardinality k such that the graph G − V k is a tree but for any subset V k − 1 ⊂ V G of cardinality k − 1 , the graph G − V k − 1 is not a tree. In this work, we find a sharp lower and some sharp upper bounds for this new sum-connectivity index.

scholarly journals Some Properties on Estrada Index of Folded Hypercubes Networks

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Jia-Bao Liu ◽  
Xiang-Feng Pan ◽  
Jinde Cao
Keyword(s):  
Adjacency Matrix ◽  
Upper Bounds ◽  
Simple Graph ◽  
Spectral Graph Theory ◽  
Lower And Upper Bounds ◽  
Estrada Index ◽  
Spectral Graph ◽  
Explicit Formulae ◽  
Folded Hypercube ◽  
Hypercube Networks

LetGbe a simple graph withnvertices and letλ1,λ2,…,λnbe the eigenvalues of its adjacency matrix; the Estrada indexEEGof the graphGis defined as the sum of the termseλi,  i=1,2,…,n. Then-dimensional folded hypercube networksFQnare an important and attractive variant of then-dimensional hypercube networksQn, which are obtained fromQnby adding an edge between any pair of vertices complementary edges. In this paper, we establish the explicit formulae for calculating the Estrada index of the folded hypercubes networksFQnby deducing the characteristic polynomial of the adjacency matrix in spectral graph theory. Moreover, some lower and upper bounds for the Estrada index of the folded hypercubes networksFQnare proposed.

scholarly journals Computing the (k-)monopoly number of direct product of graphs

Filomat ◽  
2015 ◽  
Vol 29 (5) ◽  
pp. 1163-1171
Author(s):  
Dorota Kuziak ◽  
Iztok Peterin ◽  
Ismael Yero
Keyword(s):  
Direct Product ◽  
Minimum Degree ◽  
Upper Bounds ◽  
Simple Graph ◽  
Lower And Upper Bounds ◽  
Minimum Cardinality ◽  
Product Graphs ◽  
Product Of Graphs

Let G = (V,E) be a simple graph without isolated vertices and minimum degree ?(G), and let k ? {1-??(G)/2? ,..., ?(G)/2c?} be an integer. Given a set M ? V, a vertex v of G is said to be k-controlled by M if ?M(v)? ?(v)/2 + k where ?M(v) represents the quantity of neighbors v has in M and ?(v) the degree of v. The set M is called a k-monopoly if it k-controls every vertex v of G. The minimum cardinality of any k-monopoly is the k-monopoly number of G. In this article we study the k-monopoly number of direct product graphs. Specifically we obtain tight lower and upper bounds for the k-monopoly number of direct product graphs in terms of the k-monopoly numbers of its factors. Moreover, we compute the exact value for the k-monopoly number of several families of direct product graphs.

scholarly journals Sharp Upper and Lower Bounds of VDB Topological Indices of Digraphs

Symmetry ◽  
2021 ◽  
Vol 13 (10) ◽  
pp. 1903
Author(s):  
Juan Monsalve ◽  
Juan Rada
Keyword(s):  
Lower Bounds ◽  
Topological Index ◽  
General Setting ◽  
Upper Bounds ◽  
Topological Indices ◽  
Vertex Degree ◽  
Upper And Lower Bounds ◽  
Lower And Upper Bounds ◽  
Extremal Values

A vertex-degree-based (VDB, for short) topological index φ induced by the numbers φij was recently defined for a digraph D, as φD=12∑uvφdu+dv−, where du+ denotes the out-degree of the vertex u,dv− denotes the in-degree of the vertex v, and the sum runs over the set of arcs uv of D. This definition generalizes the concept of a VDB topological index of a graph. In a general setting, we find sharp lower and upper bounds of a symmetric VDB topological index over Dn, the set of all digraphs with n non-isolated vertices. Applications to well-known topological indices are deduced. We also determine extremal values of symmetric VDB topological indices over OTn and OG, the set of oriented trees with n vertices, and the set of all orientations of a fixed graph G, respectively.

scholarly journals Reciprocal product-degree distance of graphs

Filomat ◽  
2016 ◽  
Vol 30 (8) ◽  
pp. 2217-2231
Author(s):  
Guifu Su ◽  
Liming Xiong ◽  
Ivan Gutman ◽  
Lan Xu
Keyword(s):  
Connected Graph ◽  
Upper Bounds ◽  
Graph Invariant ◽  
Lower And Upper Bounds ◽  
Harary Index ◽  
Underlying Graph ◽  
Degree Distance ◽  
Degree Modification ◽  
Type Relation

We investigate a new graph invariant named reciprocal product-degree distance, defined as: RDD* = ?{u,v}?V(G)u?v deg(u)?deg(v)/dist(u,v) where deg(v) is the degree of the vertex v, and dist(u,v) is the distance between the vertices u and v in the underlying graph. RDD* is a product-degree modification of the Harary index. We determine the connected graph of given order with maximum RDD*-value, and establish lower and upper bounds for RDD*. Also a Nordhaus-Gaddum-type relation for RDD* is obtained.

On Energy and Laplacian Energy of Graphs

2016 ◽  
Vol 31 ◽  
pp. 167-186 ◽  
Author(s):  
Kinkar Das ◽  
Seyed Ahmad Mojalal
Keyword(s):  
Maximum Degree ◽  
Upper Bounds ◽  
Simple Graph ◽  
Lower And Upper Bounds ◽  
First Zagreb Index ◽  
Zagreb Index ◽  
Laplacian Eigenvalues ◽  
Laplacian Energy ◽  
The Absolute ◽  
The First Zagreb Index

Let $G=(V,E)$ be a simple graph of order $n$ with $m$ edges. The energy of a graph $G$, denoted by $\mathcal{E}(G)$, is defined as the sum of the absolute values of all eigenvalues of $G$. The Laplacian energy of the graph $G$ is defined as \[ LE = LE(G)=\sum^n_{i=1}\left|\mu_i-\frac{2m}{n}\right| \] where $\mu_1,\,\mu_2,\,\ldots,\,\mu_{n-1},\,\mu_n=0$ are the Laplacian eigenvalues of graph $G$. In this paper, some lower and upper bounds for $\mathcal{E}(G)$ are presented in terms of number of vertices, number of edges, maximum degree and the first Zagreb index, etc. Moreover, a relation between energy and Laplacian energy of graphs is given.

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