Related papers: The edge-girth-regularity of Wenger graphs

Let $F$ and $G$ be simple finite oriented graphs (without symmetric arcs). A graph $G$ is called $F$-irregular if any two distinct vertices in $G$ belong to a different number of subgraphs of $G$ isomorphic to $F$. In this paper, we…

Let $q=p^e$, where $p$ is a prime and $e\geq 1$ is an integer. For $m\geq 1$, let $P$ and $L$ be two copies of the $(m+1)$-dimensional vector spaces over the finite field $\mathbb{F}_q$. Consider the bipartite graph $W_m(q)$ with partite…

Confirming a conjecture of Elphick and Edwards and strengthening a spectral theorem of Wilf, Nikiforov proved that for any $K_{r+1}$-free graph $G$, $\lambda(G)^2 \leq 2 (1 - 1/r) m$, where $\lambda(G)$ is the spectral radius of $G$, and…

For a simple graph $G$, let $n$ and $m$ denote the number of vertices and edges in $G$, respectively. The Erd\H{o}s-Gallai theorem for paths states that in a simple $P_k$-free graph, $m \leq \frac{n(k-1)}{2}$, where $P_k$ denotes a path…

The resolvent energy of a graph $G$ of order $n$ is defined as $ER=\sum_{i=1}^n (n-\lambda_i)^{-1}$, where $\lambda_1,\lambda_2,\ldots,\lambda_n$ are the eigenvalues of $G$. In a recent work [Gutman et al., {\it MATCH Commun. Math. Comput.…

The edges surrounding a face of a map $M$ form a cycle $C$, called the boundary cycle of the face, and $C$ is often not a simple cycle. If the map $M$ is arc-transitive, then there is a cyclic subgroup of automorphisms of $M$ which leaves…

We determine the maximum number of edges of an $n$-vertex graph $G$ with the property that none of its $r$-cliques intersects a fixed set $M\subset V(G)$. For $(r-1)|M|\ge n$, the $(r-1)$-partite Turan graph turns out to be the unique…

Let G be a finite graph with the non-k-order property (essentially, a uniform finite bound on the size of an induced sub-half-graph). A major result of the paper applies model-theoretic arguments to obtain a stronger version of…

The Fibonacci cube of dimension n, denoted as $\Gamma\_n$, is the subgraph of the hypercube induced by vertices with no consecutive 1's. The irregularity of a graph G is the sum of |d(x)-d(y)| over all edges {x,y} of G. In two recent paper…

The following sharpening of Tur\'an's theorem is proved. Let $T_{n,p}$ denote the complete $p$--partite graph of order $n$ having the maximum number of edges. If $G$ is an $n$-vertex $K_{p+1}$-free graph with $e(T_{n,p})-t$ edges then there…

Let $\Gamma=(G,\sigma)$ be a signed graph and $A(G,\sigma)$ be its adjacency matrix. Denote by $gr(G)$ the girth of $G$, which is the length of the shortest cycle in $G$. Let $r(G,\sigma)$ be the rank of $(G,\sigma)$. In this paper, we will…

The cube Q is the usual 8-vertex graph with 12 edges. Here we give a new proof for a theorem of Erd\H{o}s and Simonovits concerning the Tur\'an number of the cube. Namely, it is shown that e(G) < n^{8/5}+(2n)^{3/2} holds for any n-vertex…

Let $H$ and $G$ be graphs such that $H$ has at least 3 vertices and is connected. The $H$-line graph of $G$, denoted by $HL(G)$, is that graph whose vertices are the edges of $G$ and where two vertices of $HL(G)$ are adjacent if they are…

Let $D=(G,\mathcal{O},w)$ be a weighted oriented graph whose edge ideal is $I(D)$. In this paper, we characterize the unmixed property of $I(D)$ for each one of the following cases: $G$ is an $SCQ$ graph; $G$ is a chordal graph; $G$ is a…

Highly regular graphs for which not all regularities are explainable by symmetries are fascinating creatures. Some of them like, e.g., the line graph of W.~Kantor's non-classical $\mathrm{GQ}(5^2,5)$, are stumbling stones for existing…

In this paper we investigate orders, longest cycles and the number of cycles of automorphisms of finite vertex-transitive graphs. In particular, we show that the order of every automorphism of a connected vertex-transitive graph with $n$…

We study a Tur\'an-type problem on edge-colored complete graphs. We show that for any $r$ and $t$, any sufficiently large $r$-edge-colored complete graph on $n$ vertices with $\Omega(n^{2-1/tr^r})$ edges in each color contains a member from…

Burr and Erd\H{o}s conjectured in 1976 that for every two integers $k>\ell\geqslant 0$ satisfying that $k\mathbb{Z}+\ell$ contains an even integer, an $n$-vertex graph containing no cycles of length $\ell$ modulo $k$ can contain at most a…

The Weisfeiler-Leman (WL) dimension of a graph is a measure for the inherent descriptive complexity of the graph. While originally derived from a combinatorial graph isomorphism test called the Weisfeiler-Leman algorithm, the WL dimension…

Let $G$ be a graph with edge set $E(G)$. Denote by $d_w$ the degree of a vertex $w$ of $G$. The sigma index of $G$ is defined as $\sum_{uv\in E(G)}(d_u-d_v)^2$. A connected graph of order $n$ and size $n+k-1$ is known as a connected…