scholarly journals Цветовая энергия некоторых кластерных графов

2021 ◽  
pp. 51-64
Author(s):  
S. D'Souza ◽  
K.P. Girija ◽  
H.J. Gowtham
Keyword(s):  
Quantum Chemistry ◽  
Adjacency Matrix ◽  
Connected Graph ◽  
Complete Graphs ◽  
Vertex Deletion ◽  
Minimum Number ◽  
Complete Bipartite Graphs ◽  
Proper Generalization ◽  
Simple Connected Graph ◽  
Cluster Graphs

Let $G$ be a simple connected graph. The energy of a graph $G$ is defined as sum of the absolute eigenvalues of an adjacency matrix of the graph $G$. It represents a proper generalization of a formula valid for the total $\pi$-electron energy of a conjugated hydrocarbon as calculated by the Huckel molecular orbital (HMO) method in quantum chemistry. A coloring of a graph $G$ is a coloring of its vertices such that no two adjacent vertices share the same color. The minimum number of colors needed for the coloring of a graph $G$ is called the chromatic number of $G$ and is denoted by $\chi(G)$. The color energy of a graph $G$ is defined as the sum of absolute values of the color eigenvalues of $G$. The graphs with large number of edges are referred as cluster graphs. Cluster graphs are graphs obtained from complete graphs by deleting few edges according to some criteria. It can be obtained on deleting some edges incident on a vertex, deletion of independent edges/triangles/cliques/path P3 etc. Bipartite cluster graphs are obtained by deleting few edges from complete bipartite graphs according to some rule. In this paper, the color energy of cluster graphs and bipartite cluster graphs are studied.

ON MOSTAR INDEX OF GRAPHS

2021 ◽  
Vol 10 (4) ◽  
pp. 2115-2129
Author(s):  
P. Kandan ◽  
S. Subramanian
Keyword(s):  
Connected Graph ◽  
Cartesian Product ◽  
Bipartite Graphs ◽  
Topological Indices ◽  
Great Success ◽  
Complete Graphs ◽  
Molecular Graphs ◽  
Graphs And Networks ◽  
Complete Bipartite Graphs ◽  
Simple Connected Graph

On the great success of bond-additive topological indices like Szeged, Padmakar-Ivan, Zagreb, and irregularity measures, yet another index, the Mostar index, has been introduced recently as a peripherality measure in molecular graphs and networks. For a connected graph G, the Mostar index is defined as $$M_{o}(G)=\displaystyle{\sum\limits_{e=gh\epsilon E(G)}}C(gh),$$ where $C(gh) \,=\,\left|n_{g}(e)-n_{h}(e)\right|$ be the contribution of edge $uv$ and $n_{g}(e)$ denotes the number of vertices of $G$ lying closer to vertex $g$ than to vertex $h$ ($n_{h}(e)$ define similarly). In this paper, we prove a general form of the results obtained by $Do\check{s}li\acute{c}$ et al.\cite{18} for compute the Mostar index to the Cartesian product of two simple connected graph. Using this result, we have derived the Cartesian product of paths, cycles, complete bipartite graphs, complete graphs and to some molecular graphs.

scholarly journals Some New Results on Various Graph Energies of the Splitting Graph

2019 ◽  
Vol 2019 ◽  
pp. 1-12
Author(s):  
Zheng-Qing Chu ◽  
Saima Nazeer ◽  
Tariq Javed Zia ◽  
Imran Ahmed ◽  
Sana Shahid
Keyword(s):  
Adjacency Matrix ◽  
Connected Graph ◽  
Absolute Value ◽  
Regular Splitting ◽  
The Absolute ◽  
Splitting Graph ◽  
Simple Connected Graph

The energy of a simple connected graph G is equal to the sum of the absolute value of eigenvalues of the graph G where the eigenvalue of a graph G is the eigenvalue of its adjacency matrix AG. Ultimately, scores of various graph energies have been originated. It has been shown in this paper that the different graph energies of the regular splitting graph S′G is a multiple of corresponding energy of a given graph G.

scholarly journals Distinct Distances in Graph Drawings

10.37236/831 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Paz Carmi ◽  
Vida Dujmović ◽  
Pat Morin ◽  
David R. Wood
Keyword(s):  
Maximum Degree ◽  
Combinatorial Geometry ◽  
Complete Graphs ◽  
Bounded Degree ◽  
Cartesian Products ◽  
Bounded Treewidth ◽  
Line Drawings ◽  
Straight Line ◽  
Minimum Number ◽  
Complete Bipartite Graphs

The distance-number of a graph $G$ is the minimum number of distinct edge-lengths over all straight-line drawings of $G$ in the plane. This definition generalises many well-known concepts in combinatorial geometry. We consider the distance-number of trees, graphs with no $K^-_4$-minor, complete bipartite graphs, complete graphs, and cartesian products. Our main results concern the distance-number of graphs with bounded degree. We prove that $n$-vertex graphs with bounded maximum degree and bounded treewidth have distance-number in ${\cal O}(\log n)$. To conclude such a logarithmic upper bound, both the degree and the treewidth need to be bounded. In particular, we construct graphs with treewidth $2$ and polynomial distance-number. Similarly, we prove that there exist graphs with maximum degree $5$ and arbitrarily large distance-number. Moreover, as $\Delta$ increases the existential lower bound on the distance-number of $\Delta$-regular graphs tends to $\Omega(n^{0.864138})$.

scholarly journals On the Graceful Game

2020 ◽  
Vol 18 (3) ◽  
Author(s):  
Atilio Luiz ◽  
Simone Dantas ◽  
Luisa Ricardo
Keyword(s):  
Connected Graph ◽  
Bipartite Graphs ◽  
Absolute Difference ◽  
Complete Graphs ◽  
Graceful Labeling ◽  
Graph Games ◽  
First Results ◽  
Winning Strategies ◽  
Complete Bipartite Graphs ◽  
The Absolute

A graceful labeling of a graph G with m edges consists in labeling the vertices of G with distinct integers from 0 to m such that, when each edge is assigned the absolute difference of the labels of its endpoints, all induced edge labels are distinct. Rosa established two well known conjectures: all trees are graceful (1966) and all triangular cacti are graceful (1988). In order to contribute to both conjectures we study these problems in the context of graph games. The graceful game was introduced by Tuza in 2017 as a two-players game on a connected graph in which the players Alice and Bob take turns labeling the vertices with distinct integers from 0 to m. Alice’s goal is to gracefully label the graph as Bob’s goal is to prevent it from happening. In this work, we present the first results in this area by showing winning strategies for Alice and Bob in complete graphs, paths, cycles, complete bipartite graphs, caterpillars, prisms, wheels, helms, webs, gear graphs, hypercubes and some powers of paths.

Tetracyclic graphs with maximal Estrada index

2017 ◽  
Vol 09 (03) ◽  
pp. 1750041 ◽  
Author(s):  
Nader Jafari Rad ◽  
Akbar Jahanbani ◽  
Doost Ali Mojdeh
Keyword(s):  
Adjacency Matrix ◽  
Connected Graph ◽  
Estrada Index ◽  
Simple Connected Graph

The Estrada index of a simple connected graph [Formula: see text] of order [Formula: see text] is defined as [Formula: see text], where [Formula: see text] are the eigenvalues of the adjacency matrix of [Formula: see text]. In this paper, we characterize all tetracyclic graphs of order [Formula: see text] with maximal Estrada index.

scholarly journals ON METRIC DIMENSION OF FUNCTIGRAPHS

2013 ◽  
Vol 05 (04) ◽  
pp. 1250060 ◽  
Author(s):  
LINDA EROH ◽  
CONG X. KANG ◽  
EUNJEONG YI
Keyword(s):  
Connected Graph ◽  
Metric Dimension ◽  
Complete Graphs ◽  
Minimum Number ◽  
Vertex Set

The metric dimension of a graph G, denoted by dim (G), is the minimum number of vertices such that each vertex is uniquely determined by its distances to the chosen vertices. Let G1and G2be disjoint copies of a graph G and let f : V(G1) → V(G2) be a function. Then a functigraphC(G, f) = (V, E) has the vertex set V = V(G1) ∪ V(G2) and the edge set E = E(G1) ∪ E(G2) ∪ {uv | v = f(u)}. We study how metric dimension behaves in passing from G to C(G, f) by first showing that 2 ≤ dim (C(G, f)) ≤ 2n - 3, if G is a connected graph of order n ≥ 3 and f is any function. We further investigate the metric dimension of functigraphs on complete graphs and on cycles.

scholarly journals On S Degrees of Vertices and S Indices of Graphs

2017 ◽  
Author(s):  
Süleyman Ediz
Keyword(s):  
Connected Graph ◽  
Chemical Properties ◽  
Topological Indices ◽  
Complete Graphs ◽  
Structure Property ◽  
Connected Graphs ◽  
Structure Activity ◽  
Structure Property Relationship ◽  
Simple Connected Graph ◽  
And Chemical Properties

Topological indices have been used to modeling biological and chemical properties of molecules in quantitive structure property relationship studies and quantitive structure activity studies. All the degree based topological indices have been defined via classical degree concept. In this paper we define a novel degree concept for a vertex of a simple connected graph: S degree. And also we define S indices of a simple connected graph by using the S degree concept. We compute the S indices for well-known simple connected graphs such as paths, stars, complete graphs and cycles.

scholarly journals New bounds on the rainbow domination subdivision number

Filomat ◽  
2014 ◽  
Vol 28 (3) ◽  
pp. 615-622 ◽  
Author(s):  
Mohyedin Falahat ◽  
Seyed Sheikholeslami ◽  
Lutz Volkmann
Keyword(s):  
Connected Graph ◽  
Open Neighborhood ◽  
Minimum Weight ◽  
Domination Number ◽  
Rainbow Domination ◽  
Minimum Number ◽  
Vertex Set ◽  
Simple Connected Graph ◽  
Domination Subdivision Number ◽  
Dominating Function

A 2-rainbow dominating function (2RDF) of a graph G is a function f from the vertex set V(G) to the set of all subsets of the set {1,2} such that for any vertex v ? V(G) with f (v) = ? the condition Uu?N(v) f(u)= {1,2} is fulfilled, where N(v) is the open neighborhood of v. The weight of a 2RDF f is the value ?(f) = ?v?V |f(v)|. The 2-rainbow domination number of a graph G, denoted by r2(G), is the minimum weight of a 2RDF of G. The 2-rainbow domination subdivision number sd?r2(G) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the 2-rainbow domination number. In this paper we prove that for every simple connected graph G of order n ? 3, sd?r2(G)? 3 + min{d2(v)|v?V and d(v)?2} where d2(v) is the number of vertices of G at distance 2 from v.

Neighbor Rupture Degree of Transformation Graphs Gxy−

2017 ◽  
Vol 28 (04) ◽  
pp. 335-355
Author(s):  
Goksen Bacak-Turan ◽  
Ekrem Oz
Keyword(s):  
Connected Graph ◽  
Bipartite Graphs ◽  
Connected Components ◽  
Complete Graphs ◽  
Maximum Order ◽  
Wheel Graphs ◽  
Complete Bipartite Graphs

A vulnerability parameter the neighbor rupture degree can be used to obtain the vulnerability of a spy network. The neighbor rupture degree of a noncomplete connected graph [Formula: see text] is defined to be [Formula: see text] where [Formula: see text] is any vertex subversion strategy of [Formula: see text], [Formula: see text] is the number of connected components in [Formula: see text], and [Formula: see text] is the maximum order of the components of [Formula: see text]. In this study, the neighbor rupture degree of transformation graphs [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] of path graphs, cycle graphs, wheel graphs, complete graphs and complete bipartite graphs are obtained.

The general position problem and strong resolving graphs

2019 ◽  
Vol 17 (1) ◽  
pp. 1126-1135 ◽  
Author(s):  
Sandi Klavžar ◽  
Ismael G. Yero
Keyword(s):  
Connected Graph ◽  
General Position ◽  
Clique Number ◽  
Bipartite Graphs ◽  
Complete Graphs ◽  
Direct Products ◽  
Strong Product ◽  
Product Graphs ◽  
Complete Bipartite Graphs ◽  
Position Problem

Abstract The general position number gp(G) of a connected graph G is the cardinality of a largest set S of vertices such that no three pairwise distinct vertices from S lie on a common geodesic. It is proved that gp(G) ≥ ω(GSR), where GSR is the strong resolving graph of G, and ω(GSR) is its clique number. That the bound is sharp is demonstrated with numerous constructions including for instance direct products of complete graphs and different families of strong products, of generalized lexicographic products, and of rooted product graphs. For the strong product it is proved that gp(G ⊠ H) ≥ gp(G)gp(H), and asked whether the equality holds for arbitrary connected graphs G and H. It is proved that the answer is in particular positive for strong products with a complete factor, for strong products of complete bipartite graphs, and for certain strong cylinders.

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