scholarly journals On a Conjecture about the Sombor Index of Graphs

Symmetry ◽  
2021 ◽  
Vol 13 (10) ◽  
pp. 1830
Author(s):  
Kinkar Chandra Das ◽  
Ali Ghalavand ◽  
Ali Reza Ashrafi
Keyword(s):  
Graph Invariant ◽  
Graph Invariants ◽  
Zagreb Indices ◽  
Pendent Vertex ◽  
Vertex Set ◽  
Cyclic Graphs ◽  
Pendent Vertices

Let G be a graph with vertex set V(G) and edge set E(G). A graph invariant for G is a number related to the structure of G which is invariant under the symmetry of G. The Sombor and reduced Sombor indices of G are two new graph invariants defined as SO(G)=∑uv∈E(G)dG(u)2+dG(v)2 and SOred(G)=∑uv∈E(G)dG(u)−12+dG(v)−12, respectively, where dG(v) is the degree of the vertex v in G. We denote by Hn,ν the graph constructed from the star Sn by adding ν edge(s), 0≤ν≤n−2, between a fixed pendent vertex and ν other pendent vertices. Réti et al. [T. Réti, T Došlić and A. Ali, On the Sombor index of graphs, Contrib. Math.3 (2021) 11–18] proposed a conjecture that the graph Hn,ν has the maximum Sombor index among all connected ν-cyclic graphs of order n, where 0≤ν≤n−2. In some earlier works, the validity of this conjecture was proved for ν≤5. In this paper, we confirm that this conjecture is true, when ν=6. The Sombor index in the case that the number of pendent vertices is less than or equal to n−ν−2 is investigated, and the same results are obtained for the reduced Sombor index. Some relationships between Sombor, reduced Sombor, and first Zagreb indices of graphs are also obtained.

scholarly journals NON-CYCLIC GRAPH ASSOCIATED WITH A GROUP

2009 ◽  
Vol 08 (02) ◽  
pp. 243-257 ◽  
Author(s):  
A. ABDOLLAHI ◽  
A. MOHAMMADI HASSANABADI
Keyword(s):  
Solvable Group ◽  
Cyclic Group ◽  
Cyclic Subgroup ◽  
Maximum Size ◽  
Clique Number ◽  
Simple Graph ◽  
Complete Subgraph ◽  
Vertex Set ◽  
Cyclic Graphs ◽  
Cyclic Graph

We associate a graph [Formula: see text] to a non locally cyclic group G (called the non-cyclic graph of G) as follows: take G\ Cyc (G) as vertex set, where Cyc (G) = {x ∈ G | 〈x,y〉 is cyclic for all y ∈ G} is called the cyclicizer of G, and join two vertices if they do not generate a cyclic subgroup. For a simple graph Γ, w(Γ) denotes the clique number of Γ, which is the maximum size (if it exists) of a complete subgraph of Γ. In this paper we characterize groups whose non-cyclic graphs have clique numbers at most 4. We prove that a non-cyclic group G is solvable whenever [Formula: see text] and the equality for a non-solvable group G holds if and only if G/ Cyc (G) ≅ A5 or S5.

scholarly journals Introducing New Exponential Zagreb Indices for Graphs

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Nihat Akgunes ◽  
Busra Aydin
Keyword(s):  
Graph Invariants ◽  
Zagreb Index ◽  
Zagreb Indices ◽  
Graph Operations ◽  
General Graphs

New graph invariants, named exponential Zagreb indices, are introduced for more than one type of Zagreb index. After that, in terms of exponential Zagreb indices, lists on equality results over special graphs are presented as well as some new bounds on unicyclic, acyclic, and general graphs are obtained. Moreover, these new graph invariants are determined for some graph operations.

A novel graph invariant: The third leap Zagreb index under several graph operations

2019 ◽  
Vol 11 (05) ◽  
pp. 1950054 ◽  
Author(s):  
Durbar Maji ◽  
Ganesh Ghorai
Keyword(s):  
Tensor Product ◽  
Cartesian Product ◽  
Lexicographic Product ◽  
Graph Invariant ◽  
Zagreb Index ◽  
Zagreb Indices ◽  
The Third ◽  
Graph Operations ◽  
Strong Product ◽  
Corona Product

The third leap Zagreb index of a graph [Formula: see text] is denoted as [Formula: see text] and is defined as [Formula: see text], where [Formula: see text] and [Formula: see text] are the 2-distance degree and the degree of the vertex [Formula: see text] in [Formula: see text], respectively. The first, second and third leap Zagreb indices were introduced by Naji et al. [A. M. Naji, N. D. Soner and I. Gutman, On leap Zagreb indices of graphs, Commun. Combin. Optim. 2(2) (2017) 99–117] in 2017. In this paper, the behavior of the third leap Zagreb index under several graph operations like the Cartesian product, Corona product, neighborhood Corona product, lexicographic product, strong product, tensor product, symmetric difference and disjunction of two graphs is studied.

Product version of reciprocal Gutman indices of composite graphs

2017 ◽  
Vol 26 (2) ◽  
pp. 211-219
Author(s):  
K. Pattabiraman
Keyword(s):  
Tensor Product ◽  
Upper Bounds ◽  
Graph Invariants ◽  
Harary Index ◽  
Connected Graphs ◽  
Zagreb Indices ◽  
Strong Product

In this paper, we present the upper bounds for the product version of reciprocal Gutman indices of the tensor product, join and strong product of two connected graphs in terms of other graph invariants including the Harary index and Zagreb indices.

scholarly journals Zagreb, multiplicative zagreb indices and coindices of 〖nc〗_n (k) and 〖ca〗_3 (c_6 ) graphs

2018 ◽  
Vol 36 (2) ◽  
pp. 9-15
Author(s):  
Vida Ahmadi ◽  
Mohammad Reza Darafshe
Keyword(s):  
Connected Graph ◽  
Zagreb Indices ◽  
The Third ◽  
Degree Of Vertex ◽  
Vertex Set ◽  
Simple Connected Graph

Let  be a simple connected graph with vertex set V and edge set E. The first, second and third Zagreb indices of G are defind, respectivly by: ,   and   where  is the degree of vertex u in G and uv is an edge of G, connecting the vertices u and v. Recently, the first and second multiplicative Zagreb indices of graph  are defind by:  and . The first and second Zagreb coindices of graph are defind by:  and .  and , named as multiplicative Zagreb coindices. In this article, we compute the first, second and the third Zagreb indices and the first and second multiplicative Zagreb indices of some  graphs. The first and second Zagreb coindices and the first and second multiplicative Zagreb coindices of these graphs are also computed.

scholarly journals Application of some graph invariants to the analysis of multiprocessor interconnection networks

2008 ◽  
Vol 18 (2) ◽  
pp. 173-186 ◽  
Author(s):  
Dragos Cvetkovic ◽  
Tatjana Davidovic
Keyword(s):  
Interconnection Networks ◽  
Recent Literature ◽  
Vertex Degree ◽  
Graph Invariant ◽  
Graph Invariants ◽  
Connected Graphs ◽  
Largest Eigenvalue ◽  
Multiprocessor Interconnection Networks ◽  
Related Graph ◽  
Multiprocessor Interconnection

Let G be a graph with diameter D, maximum vertex degree ?, the largest eigenvalue ?1 and m distinct eigenvalues. The products m? and (D+1) ?1 are called the tightness of G of the first and second type, respectively. In the recent literature it was suggested that graphs with a small tightness of the first type are good models for the multiprocessor interconnection networks. We study these and some other types of tightness and some related graph invariants and demonstrate their usefulness in the analysis of multiprocessor interconnection networks. Tightness values for graphs of some standard interconnection networks are determined. We also present some facts showing that the tightness of the second type is a relevant graph invariant. We prove that the number of connected graphs with a bounded tightness is finite.

scholarly journals Some properties of the Zagreb indices

Filomat ◽  
2018 ◽  
Vol 32 (7) ◽  
pp. 2667-2675
Author(s):  
Emina Milovanovic ◽  
Igor Milovanovic ◽  
Muhammad Jamil
Keyword(s):  
Spectral Radius ◽  
Lower Bounds ◽  
Simple Graph ◽  
Graph Invariants ◽  
Zagreb Indices

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 G are adjacent, it is denoted as i ~ j. Graph invariants referred to as the first, second and the first reformulated Zagreb indices are defined as M1=?ni=1 di2, M2= ?i~j didj and EM1 = ?mi=1 d(ei)2, respectively. Let ?1 ? ?2? ... ?n be eigenvalues of G. With ?(G) = ?1 a spectral radius of G is denoted. Lower bounds for invariants M1, M2, EM1 and ?(G) are obtained.

scholarly journals On Realization and Characterization of Topological Indices

2019 ◽  
Vol 9 (1) ◽  
pp. 715-718
Keyword(s):  
Topological Index ◽  
Fixed Number ◽  
Topological Indices ◽  
Molecular Structures ◽  
Real Numbers ◽  
Graph Invariants ◽  
Chemical Graph ◽  
Zagreb Indices ◽  
Chemical Graph Theory

In chemical graph theory, topological index is one of the graph invariants which is a fixed number based on structure of a graph. Topological index is used as one of the tool to analyze molecular structures and for proper and optimal design of nanostructure. In this paper we realize the real numbers that are topological indices such as Zagreb indices, Randic index, NK-index, multiplicative F-index and multiplicative Zagreb indices along with some characterizations.

scholarly journals On Face Index of Silicon Carbides

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Xiujun Zhang ◽  
Ali Raza ◽  
Asfand Fahad ◽  
Muhammad Kamran Jamil ◽  
Muhammad Anwar Chaudhry ◽  
...  
Keyword(s):  
Physical Properties ◽  
Computer Networks ◽  
Chemical Structure ◽  
Topological Indices ◽  
Graph Invariant ◽  
Graph Invariants ◽  
Chemical Structures ◽  
Boiling Points ◽  
Silicon Carbides ◽  
The Face

Several graph invariants have been defined and studied, which present applications in nanochemistry, computer networks, and other areas of science. One vastly studied class of the graph invariants is the class of the topological indices, which helps in the studies of chemical, biological, and physical properties of a chemical structure. One recently introduced graph invariant is the face index, which can assist in predicting the energy and the boiling points of the certain chemical structures. In this paper, we drive the analytical closed formulas of face index of silicon carbides Si2C3−Ia,b, Si2C3−IIa,b, Si2C3−IIIa,b, and SiC3−IIIa,b.

scholarly journals The Differential on Graph Operator Q(G)

Symmetry ◽  
2020 ◽  
Vol 12 (5) ◽  
pp. 751
Author(s):  
Ludwin Basilio ◽  
Jair Simon ◽  
Jesús Leaños ◽  
Omar Cayetano
Keyword(s):  
Connected Graph ◽  
Independence Number ◽  
Vertex Cover ◽  
Domination Number ◽  
Graph Invariants ◽  
Maximum Value ◽  
Vertex Set ◽  
Simple Connected Graph

If G = ( V ( G ) , E ( G ) ) is a simple connected graph with the vertex set V ( G ) and the edge set E ( G ) , S is a subset of V ( G ) , and let B ( S ) be the set of neighbors of S in V ( G ) ∖ S . Then, the differential of S ∂ ( S ) is defined as | B ( S ) | − | S | . The differential of G, denoted by ∂ ( G ) , is the maximum value of ∂ ( S ) for all subsets S ⊆ V ( G ) . The graph operator Q ( G ) is defined as the graph that results by subdividing every edge of G once and joining pairs of these new vertices iff their corresponding edges are incident in G. In this paper, we study the relations between ∂ ( G ) and ∂ ( Q ( G ) ) . Besides, we exhibit some results relating the differential ∂ ( G ) and well-known graph invariants, such as the domination number, the independence number, and the vertex-cover number.

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