Subgraph densities in a surface

Given a fixed graph H that embeds in a surface $\Sigma$ , what is the maximum number of copies of H in an n-vertex graph G that embeds in $\Sigma$ ? We show that the answer is $\Theta(n^{f(H)})$ , where f(H) is a graph invariant called the ‘flap-number’ of H, which is independent of $\Sigma$ . This simultaneously answers two open problems posed by Eppstein ((1993) J. Graph Theory17(3) 409–416.). The same proof also answers the question for minor-closed classes. That is, if H is a $K_{3,t}$ minor-free graph, then the maximum number of copies of H in an n-vertex $K_{3,t}$ minor-free graph G is $\Theta(n^{f'(H)})$ , where f′(H) is a graph invariant closely related to the flap-number of H. Finally, when H is a complete graph we give more precise answers.

On Omega Index and Average Degree of Graphs

Average degree of a graph is defined to be a graph invariant equal to the arithmetic mean of all vertex degrees and has many applications, especially in determining the irregularity degrees of networks and social sciences. In this study, some properties of average degree have been studied. Effect of vertex deletion on this degree has been determined and a new proof of the handshaking lemma has been given. Using a recently defined graph index called o m e g a index, average degree of trees, unicyclic, bicyclic, and tricyclic graphs have been given, and these have been generalized to k -cyclic graphs. Also, the effect of edge deletion has been calculated. The average degree of some derived graphs and some graph operations have been determined.

On the r-dynamic coloring of some fan graph families

In this paper, we determine the r-dynamic chromatic number of the fan graph Fm,n and determine sharp bounds of this graph invariant for four related families of graphs: The middle graph M(Fm,n ), the total graph T (Fm,n ), the central graph C(Fm,n ) and the line graph L(Fm,n ). In addition, we determine the r-dynamic chromatic number of each one of these four families of graphs in case of being m = 1.

Mostar index and edge Mostar index of polymers

Let $$G=(V,E)$$ G = ( V , E ) be a graph and $$e=uv\in E$$ e = u v ∈ E . Define $$n_u(e,G)$$ n u ( e , G ) be the number of vertices of G closer to u than to v. The number $$n_v(e,G)$$ n v ( e , G ) can be defined in an analogous way. The Mostar index of G is a new graph invariant defined as $$Mo(G)=\sum _{uv\in E(G)}|n_u(uv,G)-n_v(uv,G)|$$ M o ( G ) = ∑ u v ∈ E ( G ) | n u ( u v , G ) - n v ( u v , G ) | . The edge version of Mostar index is defined as $$Mo_e(G)=\sum _{e=uv\in E(G)} |m_u(e|G)-m_v(G|e)|$$ M o e ( G ) = ∑ e = u v ∈ E ( G ) | m u ( e | G ) - m v ( G | e ) | , where $$m_u(e|G)$$ m u ( e | G ) and $$m_v(e|G)$$ m v ( e | G ) are the number of edges of G lying closer to vertex u than to vertex v and the number of edges of G lying closer to vertex v than to vertex u, respectively. Let G be a connected graph constructed from pairwise disjoint connected graphs $$G_1,\ldots ,G_k$$ G 1 , … , G k by selecting a vertex of $$G_1$$ G 1 , a vertex of $$G_2$$ G 2 , and identifying these two vertices. Then continue in this manner inductively. We say that G is a polymer graph, obtained by point-attaching from monomer units $$G_1,\ldots ,G_k$$ G 1 , … , G k . In this paper, we consider some particular cases of these graphs that are of importance in chemistry and study their Mostar and edge Mostar indices.

On a Conjecture about the Sombor Index of Graphs

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.

Omega Index of Line and Total Graphs

A derived graph is a graph obtained from a given graph according to some predetermined rules. Two of the most frequently used derived graphs are the line graph and the total graph. Calculating some properties of a derived graph helps to calculate the same properties of the original graph. For this reason, the relations between a graph and its derived graphs are always welcomed. A recently introduced graph index which also acts as a graph invariant called omega is used to obtain such relations for line and total graphs. As an illustrative exercise, omega values and the number of faces of the line and total graphs of some frequently used graph classes are calculated.

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

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.

Local Inclusive Distance Vertex Irregular Graphs

Let G=(V,E) be a simple graph. A vertex labeling f:V(G)→{1,2,⋯,k} is defined to be a local inclusive (respectively, non-inclusive) d-distance vertex irregular labeling of a graph G if for any two adjacent vertices x,y∈V(G) their weights are distinct, where the weight of a vertex x∈V(G) is the sum of all labels of vertices whose distance from x is at most d (respectively, at most d but at least 1). The minimum k for which there exists a local inclusive (respectively, non-inclusive) d-distance vertex irregular labeling of G is called the local inclusive (respectively, non-inclusive) d-distance vertex irregularity strength of G. In this paper, we present several basic results on the local inclusive d-distance vertex irregularity strength for d=1 and determine the precise values of the corresponding graph invariant for certain families of graphs.

An Approach to the Geometric-Arithmetic Index for Graphs under Transformations’ Fact over Pendent Paths

Graph theory is a dynamic tool for designing and modeling of an interconnection system by a graph. The vertices of such graph are processor nodes and edges are the connections between these processors nodes. The topology of a system decides its best use. Geometric-arithmetic index is one of the most studied graph invariant to characterize the topological aspects of underlying interconnection networks or graphs. Transformation over graph is also an important tool to define new network of their own choice in computer science. In this work, we discuss transformed family of graphs. Let Γ n k , l be the connected graph comprises by k number of pendent path attached with fully connected vertices of the n-vertex connected graph Γ . Let A α Γ n k , l and A α β Γ n k , l be the transformed graphs under the fact of transformations A α and A α β , 0 ≤ α ≤ l , 0 ≤ β ≤ k − 1 , respectively. In this work, we obtained new inequalities for the graph family Γ n k , l and transformed graphs A α Γ n k , l and A α β Γ n k , l which involve GA Γ . The presence of GA Γ makes the inequalities more general than all those which were previously defined for the GA index. Furthermore, we characterize extremal graphs which make the inequalities tight.

Computing the Narumi–Katayama Index and Modified Narumi–Katayama Index of Some Families of Dendrimers and Tetrathiafulvalene

A dendrimer is an artificially manufactured or synthesized molecule built up from branched units called monomers. In mathematical chemistry, a particular attention is given to degree-based graph invariant. The Narumi–Katayama index and its modified Narumi–Katayama index of a graph G denoted by NK (G) and NK ∗ (G) are equal to the product of the degrees of the vertices of G. In this paper, we calculate the Narumi–Katayama Index and modified Narumi–Katayama index for some families of dendrimers.