scholarly journals On Mostar and Edge Mostar Indices of Graphs

2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
Ali Ghalavand ◽  
Ali Reza Ashrafi ◽  
Mardjan Hakimi-Nezhaad
Keyword(s):  
Upper Bound ◽  
Graph Invariants ◽  
Open Questions ◽  
Bicyclic Graphs ◽  
Extremal Values

Let G be a graph with edge set E G and e = u v ∈ E G . Define n u e , G and m u e , G to be the number of vertices of G closer to u than to v and the number of edges of G closer to u than to v , respectively. The numbers n v e , G and m v e , G can be defined in an analogous way. The Mostar and edge Mostar indices of G are new graph invariants defined as M o G = ∑ u v ∈ E G n u u v , G − n v u v , G and M o e G = ∑ u v ∈ E G m u u v , G − m v u v , G , respectively. In this paper, an upper bound for the Mostar and edge Mostar indices of a tree in terms of its diameter is given. Next, the trees with the smallest and the largest Mostar and edge Mostar indices are also given. Finally, a recent conjecture of Liu, Song, Xiao, and Tang (2020) on bicyclic graphs with a given order, for which extremal values of the edge Mostar index are attained, will be proved. In addition, some new open questions are presented.

A Note on the Kirchhoff and Additive Degree-Kirchhoff Indices of Graphs

2015 ◽  
Vol 70 (6) ◽  
pp. 459-463 ◽  
Author(s):  
Yujun Yang ◽  
Douglas J. Klein
Keyword(s):  
Upper Bound ◽  
Extremal Graphs ◽  
Graph Invariants ◽  
Kirchhoff Index ◽  
Resistance Distance

AbstractTwo resistance-distance-based graph invariants, namely, the Kirchhoff index and the additive degree-Kirchhoff index, are studied. A relation between them is established, with inequalities for the additive degree-Kirchhoff index arising via the Kirchhoff index along with minimum, maximum, and average degrees. Bounds for the Kirchhoff and additive degree-Kirchhoff indices are also determined, and extremal graphs are characterised. In addition, an upper bound for the additive degree-Kirchhoff index is established to improve a previously known result.

scholarly journals Bicyclic graphs with extremal values of PI index

2013 ◽  
Vol 161 (3) ◽  
pp. 395-403 ◽  
Author(s):  
Žana Kovijanić Vukićević ◽  
Dragan Stevanović
Keyword(s):  
Pi Index ◽  
Bicyclic Graphs ◽  
Extremal Values

scholarly journals Locating-Total Domination Number of Cacti Graphs

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Jianxin Wei ◽  
Uzma Ahmad ◽  
Saira Hameed ◽  
Javaria Hanif
Keyword(s):  
Upper Bound ◽  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Total Domination ◽  
Total Domination Number ◽  
Unicyclic Graphs ◽  
Total Dominating Set ◽  
Bicyclic Graphs ◽  
Number Of Cycles

For a connected graph J, a subset W ⊆ V J is termed as a locating-total dominating set if for a ∈ V J ,   N a ∩ W ≠ ϕ , and for a ,   b ∈ V J − W ,   N a ∩ W ≠ N b ∩ W . The number of elements in a smallest such subset is termed as the locating-total domination number of J. In this paper, the locating-total domination number of unicyclic graphs and bicyclic graphs are studied and their bounds are presented. Then, by using these bounds, an upper bound for cacti graphs in terms of their order and number of cycles is estimated. Moreover, the exact values of this domination variant for some families of cacti graphs including tadpole graphs and rooted products are also determined.

scholarly journals Extremal Values of Variable Sum Exdeg Index for Conjugated Bicyclic Graphs

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Muhammad Rizwan ◽  
Akhlaq Ahmad Bhatti ◽  
Muhammad Javaid ◽  
Ebenezer Bonyah
Keyword(s):  
Connected Graph ◽  
Perfect Matching ◽  
Matching Number ◽  
Bicyclic Graph ◽  
Bicyclic Graphs ◽  
Extremal Values

A connected graph G V , E in which the number of edges is one more than its number of vertices is called a bicyclic graph. A perfect matching of a graph is a matching in which every vertex of the graph is incident to exactly one edge of the matching set such that the number of vertices is two times its matching number. In this paper, we investigated maximum and minimum values of variable sum exdeg index, SEI a for bicyclic graphs with perfect matching for k ≥ 5 and a > 1 .

Extremal values of energy over oriented bicyclic graphs

2019 ◽  
Vol 342 ◽  
pp. 26-34 ◽  
Author(s):  
Juan Monsalve ◽  
Juan Rada ◽  
Yongtang Shi
Keyword(s):  
Bicyclic Graphs ◽  
Extremal Values

scholarly journals The Signless Laplacian Spectral Radius of Unicyclic and Bicyclic Graphs with a Given Girth

10.37236/670 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Ke Li ◽  
Ligong Wang ◽  
Guopeng Zhao
Keyword(s):  
Spectral Radius ◽  
Upper Bound ◽  
Extremal Graph ◽  
Laplacian Spectral Radius ◽  
Unicyclic Graphs ◽  
Signless Laplacian Spectral Radius ◽  
Disjoint Cycles ◽  
Signless Laplacian ◽  
Unique Graph ◽  
Bicyclic Graphs

Let $\mathcal{U}(n,g)$ and $\mathcal{B}(n,g)$ be the set of unicyclic graphs and bicyclic graphs on $n$ vertices with girth $g$, respectively. Let $\mathcal{B}_{1}(n,g)$ be the subclass of $\mathcal{B}(n,g)$ consisting of all bicyclic graphs with two edge-disjoint cycles and $\mathcal{B}_{2}(n,g)=\mathcal{B}(n,g)\backslash\mathcal{B}_{1}(n,g)$. This paper determines the unique graph with the maximal signless Laplacian spectral radius among all graphs in $\mathcal{U}(n,g)$ and $\mathcal{B}(n,g)$, respectively. Furthermore, an upper bound of the signless Laplacian spectral radius and the extremal graph for $\mathcal{B}(n,g)$ are also given.

Extremal Values of

1998 ◽  
Vol 41 (3) ◽  
pp. 335-347 ◽  
Author(s):  
P. Codecà ◽  
M. Nair
Keyword(s):  
Upper Bound ◽  
Riemann Hypothesis ◽  
Classical Theorem ◽  
Extremal Values

AbstractThe function Δ(x, N) as defined in the title is closely associated via Δ(N) = supx |Δ(x, N)| to several problems in the upper bound sieve. It is also known via a classical theorem of Franel that certain conjectured bounds involving averages of Δ(x, N) are equivalent to the Riemann Hypothesis. We improve the unconditional bounds which have been hitherto obtained for Δ(N) and show that these are close to being optimal. Several auxiliary results relating Δ(Np) to Δ(N), where p is a prime with pN, are also obtained and two new conjectures stated.

scholarly journals Bicyclic graphs with maximum degree resistance distance

Filomat ◽  
2016 ◽  
Vol 30 (6) ◽  
pp. 1625-1632 ◽  
Author(s):  
Junfeng Du ◽  
Jianhua Tu
Keyword(s):  
Maximum Degree ◽  
Minimum Degree ◽  
Theoretical Chemistry ◽  
Graph Invariants ◽  
Resistance Distance ◽  
Unicyclic Graphs ◽  
Degree Of Vertex ◽  
Bicyclic Graphs

Graph invariants, based on the distances between the vertices of a graph, are widely used in theoretical chemistry. Recently, Gutman, Feng and Yu (Transactions on Combinatorics, 01 (2012) 27- 40) introduced the degree resistance distance of a graph G, which is defined as DR(G) = ?{u,v}?V(G)[dG(u)+dG(v)]RG(u,v), where dG(u) is the degree of vertex u of the graph G, and RG(u, v) denotes the resistance distance between the vertices u and v of the graph G. Further, they characterized n-vertex unicyclic graphs having minimum and second minimum degree resistance distance. In this paper, we characterize n-vertex bicyclic graphs having maximum degree resistance distance.

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