scholarly journals An Approach to the Extremal Inverse Degree Index for Families of Graphs with Transformation Effect

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Muhammad Asif ◽  
Muhammad Hussain ◽  
Hamad Almohamedh ◽  
Khalid M. Alhamed ◽  
Sultan Almotairi
Keyword(s):  
Computer Program ◽  
Regular Graph ◽  
Connected Graph ◽  
Topological Index ◽  
Unicyclic Graph ◽  
Fixed Length ◽  
Inverse Degree ◽  
Fully Connected

The inverse degree index is a topological index first appeared as a conjuncture made by computer program Graffiti in 1988. In this work, we use transformations over graphs and characterize the inverse degree index for these transformed families of graphs. We established bonds for different families of n -vertex connected graph with pendent paths of fixed length attached with fully connected vertices under the effect of transformations applied on these paths. Moreover, we computed exact values of the inverse degree index for regular graph specifically unicyclic graph.

Extermal forgotten topological index of quasi-unicyclic graphs

2021 ◽  
pp. 2250041
Author(s):  
Amir Taghi Karimi
Keyword(s):  
Lower Bounds ◽  
Connected Graph ◽  
Topological Index ◽  
Unicyclic Graph ◽  
Upper And Lower Bounds ◽  
Unicyclic Graphs ◽  
Degree Of A Vertex

The forgotten topological index of a graph [Formula: see text], denoted by [Formula: see text], is defined as the sum of weights [Formula: see text] overall edges [Formula: see text] of [Formula: see text], where [Formula: see text] denotes the degree of a vertex [Formula: see text]. The graph [Formula: see text] is called a quasi-unicyclic graph if there exists a vertex [Formula: see text] such that [Formula: see text] is a connected graph with a unique cycle. In this paper, we give sharp upper and lower bounds for the F-index (forgotten topological index) of the quasi-unicyclic graphs.

scholarly journals Computing Fifth Geometric-Arithmetic Index for Circumcoronene series of benzenoid Hk

2007 ◽  
Vol 3 (1) ◽  
pp. 143-148 ◽  
Author(s):  
Mohammad Reza Farahani
Keyword(s):  
Connected Graph ◽  
Topological Index ◽  
Degree Of Vertex ◽  
Simple Connected Graph ◽  
Ga Index

Let G=(V; E) be a simple connected graph. The sets of vertices and edges of G are denoted by V=V(G) and E=E (G), respectively. The geometric-arithmetic index is a topological index was introduced by Vukicevic and Furtula in 2009 and defined as  in which degree of vertex u denoted by dG(u) (or du for short). In 2011, A. Graovac et al defined a new version of GA index as  where  The goal of this paper is to compute the fifth geometric-arithmetic index for "Circumcoronene series of benzenoid Hk (k≥1)".

scholarly journals Smallest regular graphs without near 1-factors

1986 ◽  
Vol 41 (2) ◽  
pp. 193-210 ◽  
Author(s):  
Gary Chartrand ◽  
Sergio Ruiz ◽  
Curtiss E. Wall
Keyword(s):  
Regular Graph ◽  
Connected Graph ◽  
Regular Graphs ◽  
Cubic Graph ◽  
Subgraph Isomorphic ◽  
Even Order

AbstractA near 1-factor of a graph of order 2n ≧ 4 is a subgraph isomorphic to (n − 2) K2 ∪ P3 ∪ K1. Wallis determined, for each r ≥ 3, the order of a smallest r-regular graph of even order without a 1-factor; while for each r ≧ 3, Chartrand, Goldsmith and Schuster determined the order of a smallest r-regular, (r − 2)-edge-connected graph of even order without a 1-factor. These results are extended to graphs without near 1-factors. It is known that every connected, cubic graph with less than six bridges has a near 1-factor. The order of a smallest connected, cubic graph with exactly six bridges and no near 1-factor is determined.

scholarly journals THE HEIDER BALANCE: A CONTINUOUS APPROACH

2005 ◽  
Vol 16 (05) ◽  
pp. 707-716 ◽  
Author(s):  
KRZYSZTOF KUŁAKOWSKI ◽  
PRZEMYSŁAW GAWROŃSKI ◽  
PIOTR GRONEK
Keyword(s):  
Connected Graph ◽  
Social Group ◽  
Statistical Data ◽  
Structure Theorem ◽  
System Size ◽  
Random Numbers ◽  
Continuous Approach ◽  
Internal Relations ◽  
Group Members ◽  
Fully Connected

The Heider balance (HB) is investigated in a fully connected graph of N nodes. The links are described by a real symmetric array r (i, j), i, j =1, …, N. In a social group, nodes represent group members and links represent relations between them, positive (friendly) or negative (hostile). At the balanced state, r (i, j) r (j, k) r (k, i) > 0 for all the triads (i, j, k). As follows from the structure theorem of Cartwright and Harary, at this state the group is divided into two subgroups, with friendly internal relations and hostile relations between the subgroups. Here the system dynamics is proposed to be determined by a set of differential equations, [Formula: see text]. The form of equations guarantees that once HB is reached, it persists. Also, for N =3 the dynamics reproduces properly the tendency of the system to the balanced state. The equations are solved numerically. Initially, r (i, j) are random numbers distributed around zero with a symmetric uniform distribution of unit width. Calculations up to N =500 show that HB is always reached. Time τ(N) to get the balanced state varies with the system size N as N-1/2. The spectrum of relations, initially narrow, gets very wide near HB. This means that the relations are strongly polarized. In our calculations, the relations are limited to a given range around zero. With this limitation, our results can be helpful in an interpretation of some statistical data.

SUPER SPANNING CONNECTIVITY OF THE FULLY CONNECTED CUBIC NETWORKS

2010 ◽  
Vol 11 (01n02) ◽  
pp. 61-70 ◽  
Author(s):  
CHERNG CHIN ◽  
HUAI-CHIH CHEN ◽  
LIH-HSING HSU ◽  
SHANG-CHIA CHIOU ◽  
KUO-TUNG LAI
Keyword(s):  
Connected Graph ◽  
Disjoint Paths ◽  
Fully Connected

A k-containerC(u, v) of G between u and v is a set of k internally disjoint paths between u and v. A k-container C(u,v) of G is a k*-container if it contains all vertices of G. A graph G is k*-connected if there exists a k*-container between any two distinct vertices. The spanning connectivity of G, κ*(G), is defined to be the largest integer k such that G is w*-connected for all 1 ≤ w ≤ k if G is a 1*-connected graph and undefined otherwise. A graph G is super spanning connected if κ* (G) = κ(G). In this paper, we prove that the n-dimensional fully connected cubic network FCCNn is super spanning connected.

scholarly journals REMOTE SENSING SCENE CLASSIFICATION USING MULTIPLE PYRAMID POOLING

2019 ◽  
Vol XLII-2/W16 ◽  
pp. 279-284
Author(s):  
Y. Yao ◽  
H. Zhao ◽  
D. Huang ◽  
Q. Tan
Keyword(s):  
Remote Sensing ◽  
Spatial Relationship ◽  
Image Data ◽  
Remote Sensing Image ◽  
Input Image ◽  
Image Size ◽  
Scene Classification ◽  
Fixed Size ◽  
Fixed Length ◽  
Fully Connected

<p><strong>Abstract.</strong> Remote sensing image scene classification has gained remarkable attention, due to its versatile use in different applications like geospatial object detection, ground object information extraction, environment monitoring and etc. The scene not only contains the information of the ground objects, but also includes the spatial relationship between the ground objects and the environment. With rapid growth of the amount of remote sensing image data, the need for automatic annotation methods for image scenes is more urgent. This paper proposes a new framework for high resolution remote sensing images scene classification based on convolutional neural network. To eliminate the requirement of fixed-size input image, multiple pyramid pooling strategy is equipped between convolutional layers and fully connected layers. Then, the fixed-size features generated by multiple pyramid pooling layer was extended to one-dimension fixed-length vector and fed into fully connected layers. Our method could generate a fixed-length representation regardless of image size, at the same time get higher classification accuracy. On UC-Merced and NWPU-RESISC45 datasets, our framework achieved satisfying accuracies, which is 93.24% and 88.62% respectively.</p>

scholarly journals On the Resistance Matrix of a Graph

10.37236/5295 ◽  
2016 ◽  
Vol 23 (1) ◽  
Author(s):  
Jiang Zhou ◽  
Zhongyu Wang ◽  
Changjiang Bu
Keyword(s):  
Regular Graph ◽  
Connected Graph ◽  
Spanning Trees ◽  
Regular Graphs ◽  
Resistance Distance ◽  
Strongly Regular ◽  
Resistance Matrix ◽  
Number Of Spanning Trees

Let $G$ be a connected graph of order $n$. The resistance matrix of $G$ is defined as $R_G=(r_{ij}(G))_{n\times n}$, where $r_{ij}(G)$ is the resistance distance between two vertices $i$ and $j$ in $G$. Eigenvalues of $R_G$ are called R-eigenvalues of $G$. If all row sums of $R_G$ are equal, then $G$ is called resistance-regular. For any connected graph $G$, we show that $R_G$ determines the structure of $G$ up to isomorphism. Moreover, the structure of $G$ or the number of spanning trees of $G$ is determined by partial entries of $R_G$ under certain conditions. We give some characterizations of resistance-regular graphs and graphs with few distinct R-eigenvalues. For a connected regular graph $G$ with diameter at least $2$, we show that $G$ is strongly regular if and only if there exist $c_1,c_2$ such that $r_{ij}(G)=c_1$ for any adjacent vertices $i,j\in V(G)$, and $r_{ij}(G)=c_2$ for any non-adjacent vertices $i,j\in V(G)$.

scholarly journals Bounds for symmetric division deg index of graphs

Filomat ◽  
2019 ◽  
Vol 33 (3) ◽  
pp. 683-698 ◽  
Author(s):  
Kinkar Das ◽  
Marjan Matejic ◽  
Emina Milovanovic ◽  
Igor Milovanovic
Keyword(s):  
Connected Graph ◽  
Maximum Degree ◽  
Topological Index ◽  
Minimum Degree ◽  
Topological Indices ◽  
Symmetric Division ◽  
Mathematical Properties ◽  
Vertex Degrees ◽  
Additive Modeling ◽  
Simple Connected Graph

LetG = (V,E) be a simple connected graph of order n (?2) and size m, where V(G) = {1, 2,..., n}. Also let ? = d1 ? d2 ?... ? dn = ? > 0, di = d(i), be a sequence of its vertex degrees with maximum degree ? and minimum degree ?. The symmetric division deg index, SDD, was defined in [D. Vukicevic, Bond additive modeling 2. Mathematical properties of max-min rodeg index, Croat. Chem. Acta 83 (2010) 261- 273] as SDD = SDD(G) = ?i~j d2i+d2j/didj, where i~j means that vertices i and j are adjacent. In this paper we give some new bounds for this topological index. Moreover, we present a relation between topological indices of graph.

scholarly journals On the Estrada Indices of Unicyclic Graphs with Fixed Diameters

Mathematics ◽  
2021 ◽  
Vol 9 (19) ◽  
pp. 2395
Author(s):  
Wenjie Ning ◽  
Kun Wang
Keyword(s):  
Adjacency Matrix ◽  
Connected Graph ◽  
Unicyclic Graph ◽  
Unicyclic Graphs ◽  
Estrada Index

The Estrada index of a graph G is defined as EE(G)=∑i=1neλi, where λ1,λ2,…,λn are the eigenvalues of the adjacency matrix of G. A unicyclic graph is a connected graph with a unique cycle. Let U(n,d) be the set of all unicyclic graphs with n vertices and diameter d. In this paper, we give some transformations which can be used to compare the Estrada indices of two graphs. Using these transformations, we determine the graphs with the maximum Estrada indices among U(n,d). We characterize two candidate graphs with the maximum Estrada index if d is odd and three candidate graphs with the maximum Estrada index if d is even.

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