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 THE PROPORTIONAL COLORING PROBLEM: OPTIMIZING BUFFERS IN RADIO MESH NETWORKS

2012 ◽  
Vol 04 (03) ◽  
pp. 1250028 ◽  
Author(s):  
FLORIAN HUC ◽  
CLÁUDIA LINHARES SALES ◽  
HERVÉ RIVANO
Keyword(s):  
Mesh Networks ◽  
Edge Coloring ◽  
Weighted Graph ◽  
Minimum Cost ◽  
Upper Bounds ◽  
Weighted Graphs ◽  
Chromatic Index ◽  
Lower And Upper Bounds ◽  
Coloring Problem ◽  
Wireless Mesh

In this paper, we consider a new edge coloring problem to model call scheduling optimization issues in wireless mesh networks: the proportional coloring. It consists in finding a minimum cost edge coloring of a graph which preserves the proportion given by the weights associated to each of its edges. We show that deciding if a weighted graph admits a proportional coloring is pseudo-polynomial while determining its proportional chromatic index is NP-hard. We then give lower and upper bounds for this parameter that can be computed in pseudo-polynomial time. We finally identify a class of graphs and a class of weighted graphs for which the proportional chromatic index can be exactly determined.

scholarly journals Improve Some of Bounds for the Randic Estrada Index of Graphs

2020 ◽  
Author(s):  
Akbar Jahanbani
Keyword(s):  
Lower Bounds ◽  
Upper Bounds ◽  
Randić Index ◽  
Lower And Upper Bounds ◽  
Graph Invariants ◽  
Randic Index ◽  
Estrada Index ◽  
Eigenvalues Of Graphs

Let G be a graph with n vertices and let 1; 2; : : : ; n be the eigenvalues of Randic matrix. The Randic Estrada index of G is REE(G) = Ón i=1 ei . In this paper, we establish lower and upper bounds for Randic index in terms of graph invariants such as the number of vertices and eigenvalues of graphs and improve some previously published lower bounds.

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 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 Merging the Spectral Theories of Distance Estrada and Distance Signless Laplacian Estrada Indices of Graphs

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 995 ◽  
Author(s):  
Abdollah Alhevaz ◽  
Maryam Baghipur ◽  
Yilun Shang
Keyword(s):  
Distance Matrix ◽  
Wiener Index ◽  
Upper Bounds ◽  
Extremal Graphs ◽  
Lower And Upper Bounds ◽  
Graph Spectrum ◽  
Estrada Index ◽  
Signless Laplacian ◽  
Graph Parameters ◽  
Generalized Distance

Suppose that G is a simple undirected connected graph. Denote by D ( G ) the distance matrix of G and by T r ( G ) the diagonal matrix of the vertex transmissions in G, and let α ∈ [ 0 , 1 ] . The generalized distance matrix D α ( G ) is defined as D α ( G ) = α T r ( G ) + ( 1 − α ) D ( G ) , where 0 ≤ α ≤ 1 . If ∂ 1 ≥ ∂ 2 ≥ … ≥ ∂ n are the eigenvalues of D α ( G ) ; we define the generalized distance Estrada index of the graph G as D α E ( G ) = ∑ i = 1 n e ∂ i − 2 α W ( G ) n , where W ( G ) denotes for the Wiener index of G. It is clear from the definition that D 0 E ( G ) = D E E ( G ) and 2 D 1 2 E ( G ) = D Q E E ( G ) , where D E E ( G ) denotes the distance Estrada index of G and D Q E E ( G ) denotes the distance signless Laplacian Estrada index of G. This shows that the concept of generalized distance Estrada index of a graph G merges the theories of distance Estrada index and the distance signless Laplacian Estrada index. In this paper, we obtain some lower and upper bounds for the generalized distance Estrada index, in terms of various graph parameters associated with the structure of the graph G, and characterize the extremal graphs attaining these bounds. We also highlight relationship between the generalized distance Estrada index and the other graph-spectrum-based invariants, including generalized distance energy. Moreover, we have worked out some expressions for D α E ( G ) of some special classes of graphs.

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.

Extension of the Spatial PDI Model to Three Dimensions

1997 ◽  
Vol 84 (1) ◽  
pp. 176-178
Author(s):  
Frank O'Brien
Keyword(s):  
Population Density ◽  
Dimensional Space ◽  
Finite Lattice ◽  
Three Dimensional ◽  
Upper Bounds ◽  
Three Dimensions ◽  
Lower And Upper Bounds ◽  
Density Index ◽  
Three Dimensional Space

The author's population density index ( PDI) model is extended to three-dimensional distributions. A derived formula is presented that allows for the calculation of the lower and upper bounds of density in three-dimensional space for any finite lattice.

Energy of spherical fuzzy graphs

2020 ◽  
pp. 321-332
Author(s):  
S. Yahya Mohamed ◽  
A. Mohamed Ali
Keyword(s):  
Adjacency Matrix ◽  
Upper Bounds ◽  
Fuzzy Graph ◽  
Lower And Upper Bounds ◽  
Fuzzy Graphs

In this paper, the notion of energy extended to spherical fuzzy graph. The adjacency matrix of a spherical fuzzy graph is defined and we compute the energy of a spherical fuzzy graph as the sum of absolute values of eigenvalues of the adjacency matrix of the spherical fuzzy graph. Also, the lower and upper bounds for the energy of spherical fuzzy graphs are obtained.

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