Related papers: Isometric Cycles and a Generalization of Moore Gra…
The problem of finding the longest simple cycle in a directed graph is NP-hard, with critical applications in computational biology, scheduling, and network analysis. Existing approaches include exact algorithms with exponential runtimes,…
In a graph, $k$ cycles are {\em admissible} if their lengths form an arithmetic progression with common difference one or two. Let $G$ be a 2-connected graph with minimum degree at least $k\geqslant 4$. We prove that \begin{itemize} \item…
Euler graphs are characterized by the simple criterion that degree of each node is even. By restricting on the cycle types yet additional intrinsic properties of Euler graphs are unveiled. For example, regularity higher than degree two is…
This article provides sharp bounds for the maximum number of edges possible in a simple graph with restricted values of two of the three parameters, namely, maxi- mum matching size, independence number and maximum degree. We also construct…
We introduce a notion of a girth-regular graph as a $k$-regular graph for which there exists a non-descending sequence $(a_1, a_2, \dots, a_k)$ (called the signature) giving, for every vertex $u$ of the graph, the number of girth cycles the…
The dissociation number ${\rm diss}(G)$ of a graph $G$ is the maximum order of a set of vertices of $G$ inducing a subgraph that is of maximum degree at most $1$. Computing the dissociation number of a given graph is algorithmically hard…
The $\sigma$-irregularity index of a graph is defined as the sum of squared degree differences over all edges and provides a sensitive measure of structural heterogeneity. In this paper, we study the problem of maximizing $\sigma(T)$ among…
We obtain several sharp spectral bounds, approximations, and exact values for the isoperimetric number and related edge-expansion parameters of graphs. Our results focus on graph powers and on families of graphs with rich algebraic or…
The \emph{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…
In this paper we consider the degree/diameter problem, namely, given natural numbers {\Delta} \geq 2 and D \geq 1, find the maximum number N({\Delta},D) of vertices in a graph of maximum degree {\Delta} and diameter D. In this context, the…
A graph is $k$-chordal if it does not have an induced cycle with length greater than $k$. We call a graph chordal if it is $3$-chordal. Let $G$ be a graph. The distance between the vertices $x$ and $y$, denoted by $d_{G}(x,y)$, is the…
A strong edge coloring of a graph is a proper edge coloring in which every color class is an induced matching. The strong chromatic index $\chi_s'(G)$ of a graph $G$ is the minimum number of colors in a strong edge coloring of $G$. Let…
The undirected degree/diameter and degree/girth problems and their directed analogues have been studied for many decades in the search for efficient network topologies. Recently such questions have received much attention in the setting of…
The main topic considered is maximizing the number of cycles in a graph with given number of edges. In 2009, Kir\'aly conjectured that there is constant $c$ such that any graph with $m$ edges has at most $(1.4)^m$ cycles. In this paper, it…
We derive attainable upper bounds on the algebraic connectivity (spectral gap) of a regular graph in terms of its diameter and girth. This bound agrees with the well-known Alon-Boppana-Friedman bound for graphs of even diameter, but is an…
This survey on graphs of large girth consists of two parts. The first deals with some aspects of algebraic and extremal graph theory loosely related to the Moore bound. Our point of departure for the second, Ramsey theoretic, part are some…
Given a connected graph $G$, the metric (resp. edge metric) dimension of $G$ is the cardinality of the smallest ordered set of vertices that uniquely identifies every pair of distinct vertices (resp. edges) of $G$ by means of distance…
The cyclic edge-connectivity of a graph $G$ is the least $k$ such that there exists a set of $k$ edges whose removal disconnects $G$ into components where every component contains a cycle. We show that for graphs of minimum degree at least…
Let $r\geq 3$ be an integer and $G$ be a graph. Let $\delta(G), \Delta(G)$, $\alpha(G)$ and $\mu(G)$ denotes minimum degree, maximum degree, independence number and matching number of $G$, respectively. Recently, Caro, Davila and Pepper…
We consider the case in which mixed graphs (with both directed and undirected edges) are Cayley graphs of Abelian groups. In this case, some Moore bounds were derived for the maximum number of vertices that such graphs can attain. We first…