Related papers: Regularity of Powers of Unicyclic Graphs

The power graph of an arbitrary group $G$ is a simple graph with all elements of $G$ as its vertices and two vertices are adjacent if one is a positive power of another. In this paper, we generalize this concept to a graph whose vertices…

Bondy and Vince proved that a graph of minimum degree at least three contains two cycles whose lengths differ by one or two, which was conjectured by Erd\H{o}s. Gao, Li, Ma and Xie gave an average degree counterpart of Bondy-Vince's result,…

It is proved that for $n \geq 6$, the number of perfect matchings in a simple connected cubic graph on $2n$ vertices is at most $4 f_{n-1}$, with $f_n$ being the $n$-th Fibonacci number. The unique extremal graph is characterized as well.…

In this paper, we characterize the class of {\em contraction perfect} graphs which are the graphs that remain perfect after the contraction of any edge set. We prove that a graph is contraction perfect if and only if it is perfect and the…

The minimum number of distinct eigenvalues, taken over all real symmetric matrices compatible with a given graph $G$, is denoted by $q(G)$. Using other parameters related to $G$, bounds for $q(G)$ are proven and then applied to deduce…

In this paper, our goal is to characterize two graph classes based on the properties of minimal vertex (edge) separators. We first present a structural characterization of graphs in which every minimal vertex separator is a stable set. We…

We investigate the \textit{irregularity strength} ($s(G)$) and \textit{total vertex irregularity strength} ($tvs(G)$) of circulant graphs $Ci_n(1,2,...,k)$ and prove that $tvs(Ci_n(1,2,...,k))=\lceil\frac{n+2k}{2k+1}\rceil$, while…

Let $G$ be a unicyclic graph of order $n$, and let $k$ be the length of the unique cycle of $G$. For the adjacency eigenvalues of $G$, let $s^{+}(G)$ and $s^{-}(G)$ denote the sums of the squares of the positive and negative eigenvalues,…

A graph is {\em{$\ell$-holed}} if all of its induced cycles of length at least four have length exactly $\ell$. In the paper, we prove that if $G$ is an $\ell$-holed graph with odd $\ell\geq 7$, then $\chi(G)\leq {\lceil {\ell \over…

A well-known result of Verstra\"ete \cite{V00} shows that for each integer $k\geq 2$ every graph $G$ with average degree at least $8k$ contains cycles of $k$ consecutive even lengths, the shortest of which is at most twice the radius of…

A graph is regularizable if it is possible to assign weights to its edges so that all nodes have the same degree. Weights can be positive, nonnegative or arbitrary as soon as the regularization degree is not null. Positive and nonnegative…

Bicyclic graph is a connected graph in which the number of edges equals the number of vertices plus one. In this paper, we determine the graph which alone maximizes the spectral radii among all the bicyclic graphs on $n$ vertices with fixed…

A subset of vertices in a graph $G$ is considered a maximal dissociation set if it induces a subgraph with vertex degree at most 1 and it is not contained within any other dissociation sets. In this paper, it is shown that for $n\geq 3$,…

Let $G$ be a $(C_4, 2K_2)$-free graph with edge ideal $I(G)\subset \Bbbk[x_1,\dots , x_n]$. We show that $I(G)^s$ has linear resolution for every $s\geq 2$. Also, we show that every power of the vertex cover ideal of $G$ has linear…

A graph $G$ is $l$-path Hamiltonian if every path of length not exceeding $l$ is contained in a Hamiltonian cycle. It is well known that a 2-connected, $k$-regular graph $G$ on at most $3k-1$ vertices is edge-Hamiltonian if for every edge…

We prove that an inseparable graph can have any positive number of cycles with the six exceptions 2, 4, 5, 8, 9, 16, and that an inseparable cubic graph has the additional exceptions 1 and 13. The exceptions for simple inseparable cubic…

A graph $G$ is $\textit{universal}$ for a (finite) family $\mathcal{H}$ of graphs if every $H \in \mathcal{H}$ is a subgraph of $G$. For a given family $\mathcal{H}$, the goal is to determine the smallest number of edges an…

In this paper, we define (im, reg)-invariant extension of graphs and propose a new approach for Nevo and Peeva's conjecture which said that for any gap-free graph $G$ with $reg(I(G)) = 3$ and for any $k \geq 2$, $I(G)^k$ has a linear…

The stability number of a graph G, denoted by alpha(G), is the cardinality of a stable set of maximum size in G. A graph is well-covered if every maximal stable set has the same size. G is a Koenig-Egervary graph if its order equals…

The cycles are the only $2$-connected graphs in which any two nonadjacent vertices form a vertex cut. We generalize this fact by proving that for every integer $k\ge 3$ there exists a unique graph $G$ satisfying the following conditions:…