arXiv subjects

Dinesh Pandey

Publications and source records attributed to Dinesh Pandey.

10 recordsLinked to original sources

arXiv2026

On a conjecture of DeLaVi\~na and Waller

The Wiener index of a connected graph is defined as the sum of distances between all its unordered pairs of vertices. Characterising graphs on $n$ vertices with a fixed diameter that maximise the Wiener index is a long-standing open problem. This problem has been resolved fully for trees on $n$ vertices with diameter $d \in \{1,2,3,4,n-3,n-2,n-1\}$ while partial results are available for $d=5$ and $6$. In this context, a conjecture proposed by DeLaVi\~na and Waller has remained open for the last 18 years. In this paper, we establish a necessary condition for a tree to attain the maximum Wiener index among all trees on $n$ vertices with a given diameter. Using this condition, we characterise the maximal trees for diameter $n-4$ and $n-5$. Furthermore, we prove the DeLaVi\~na Waller conjecture for the classes of graphs having $0,1,2,3$ or $n-4$ cut vertices.

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arXiv2026

On structural properties of some probable $R(3, 10)$-critical graphs

The Ramsey number $R(s, t)$ is the smallest positive integer $n$ such that every graph on $n$ vertices contains either a clique of size $s$ or an independent set of size $t$. An $R(s,t)$-critical graph is a graph on $R(s,t)-1$ vertices that contains neither a clique of size $s$ nor an independent set of size $t$. It is known that $40\leq R(3, 10)\leq 42$. We study the structure of a $R(3,10)$-critical graphs by assuming $R(3, 10)=42$. We show that if such a graph exists then its minimum degree and vertex connectivity are the same and is $6, 7$ or $8$. Then we find all the possible degree sequences of such graphs. Further, we show that if such a graph exists, then its diameter is either $2$ or $3$, and if it has diameter $2$ and minimum degree $6$, then it has only $21$ choices for its degree sequence.

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arXiv2025

Extremal graphs with minimum number of connected subgraphs in a given family

The subgraph number of a vertex in a graph is defined as the number of connected subgraphs containing that vertex. The graph and its vertex which correspond to the minimum subgraph number among all graphs on $n$ vertices and $k$ cut vertices have been characterised. Further, using this characterisation, the graphs with the minimum number of connected subgraphs among all graphs on $n$ vertices and $k$ cut vertices, with girth at least $k$, have been obtained. This turns out to characterise the graphs with the minimum number of connected subgraphs among all graphs on $n$ vertices and $k$ cut vertices for $0 \leq k \leq 4$.

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arXiv2024

On vertex peripherians and Wiener index of graphs with fixed number of cut vertices

The distance of a vertex in a graph is the sum of distances from that vertex to all other vertices of the graph. The Wiener index of a graph is the sum of distances between all its unordered pairs of vertices. A graph has been obtained that contains a vertex achieving the maximum distance among all graphs on $n$ vertices with fixed number of cut vertices. Further the graphs having maximum Wiener index among all graphs on $n$ vertices with at most $3$ cut vertices have been characterised.

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arXiv2022

Total eccentricity index of graphs with fixed number of pendant or cut vertices

The total eccentricity index of a connected graph is defined as sum of the eccentricities of all its vertices. We denote the set of all connected graphs on $n$ vertices with $k$ pendant vertices by $\mathfrak{H}_{n,k}$ and denote the set of all connected graphs on $n$ vertices with $s$ cut vertices by $\mathfrak{C_{n,s}}$. In this paper, we give the sharp lower and upper bounds on the total eccentricity index over $\mathfrak{H}_{n,k}$ and the sharp lower bound for the same over $\mathfrak{C_{n,s}}$. We also provide the sharp upper bounds on the total eccentricity index over $\mathfrak{C_{n,s}}$ when $s=0,1,n-3,n-2$ and propose a problem regarding the upper bound over $\mathfrak{C_{n,s}}$ for $2\leq s\leq n-4.$

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arXiv2021

Some new central parts of connected graphs

The center, median and the security center are three central parts defined for any connected graph whereas the characteristic set, subtree core and core vertices are three central parts defined for trees only. We extend the concept of the characteristic set, subtree core and core vertices to general connected graphs and call them the characteristic center, subgraph core and core vertices, respectively. We show by examples that in a connected graph all the above six central parts can be different and also prove that for a connected vertex transitive graph each of the six central parts is the whole vertex set. Further it is shown that given any graph $G$, there exists a connected supergraph $G_{ch}$ of $G$ with the whole vertex set of $G$ as the characteristic center. Associated with the subgraph core and core vertices, we leave some unanswered question related to the graph centrality.

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arXiv2020

A conjecture on different central parts of binary trees

Let $Ω_n$ be the family of binary trees on $n$ vertices obtained by identifying the root of an rgood binary tree with a vertex of maximum eccentricity of a binary caterpillar. In the paper titled "On different middle parts of a tree (The electronic journal of combinatorics, 25 (2018), no. 3, paper 3.17, 32 pp)", Smith et al. conjectured that among all binary trees on $n$ vertices the pairwise distance between any two of center, centroid and subtree core is maximized by some member of the family $Ω_n$. We first obtain the rooted binary tree which minimizes the number of root containing subtrees and then prove this conjecture. We also obtain the binary trees which maximize these distances.

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arXiv2020

Different central parts of trees and their pairwise distances

We determine the tree which maximizes the distance between characteristic set and subtree core over all trees on $n$ vertices. The asymptotic nature of this distance is also discussed. The problem of extremizing the distance between different central parts of trees on $n$ vertices with fixed diameter is studied

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arXiv2019

Wiener index of graphs with fixed number of pendant or cut vertices

The Wiener index of a connected graph is defined as the sum of the distances between all unordered pair of its vertices. In this paper, we characterize the graphs which extremize the Wiener index among all graphs on $n$ vertices with $k$ pendant vertices. We also characterize the graph which minimizes the Wiener index over the graphs on $n$ vertices with $s$ cut vertices.

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arXiv2019

The core index of a graph

For a graph $G,$ we denote the number of connected subgraphs of $G$ by $F(G)$. For a tree $T$, $F(T)$ has been studied extensively and it has been observed that $F(T)$ has a reverse correlation with Wiener index of $T$. Based on that, we call $F(G),$ the core index of $G$. In this paper, we characterize the graphs which extremize the core index among all graphs on $n$ vertices with $k\geq 0$ connected components. We extend our study of core index to unicyclic graphs and connected graphs with fixed number of pendant vertices. We obtained the unicyclic graphs which extremize the core index over all unicyclic graphs on $n$ vertices. The graphs which extremize the core index among all unicyclic graphs with fixed girth are also obtained. Among all connected graphs on $n$ vertices with fixed number of pendant vertices, the graph which minimizes and the graph which maximizes the core index are characterized.

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