Related papers: A spectral version of the Moore problem for bipart…

Let $G$ be a $k$-degenerate graph of order $n.$ It is well-known that $G\ $has no more edges than $S_{n,k},$ the join of a complete graph of order $k$ and an independent set of order $n-k.$ In this note it is shown that $S_{n,k}$ is…

Regular and distance-regular characterizations of general graphs are well-known. In particular, the spectral excess theorem states that a connected graph G is distance-regular if and only if its spectral excess (a number that can be…

The first multiplicative Zagreb index of a graph $G$ is the product of the square of every vertex degree, while the second multiplicative Zagreb index is the product of the degree of each edge over all edges. In our work, we explore the…

A bipartite graph $G = (X \cup Y, E)$ is a 2-layer $k$-planar graph if it admits a drawing on the plane such that the vertices in $X$ and $Y$ are placed on two parallel lines respectively, edges are drawn as straight-line segments, and…

Liu, Hong, Gu, and Lai proved if the second largest eigenvalue of the adjacency matrix of graph $G$ with minimum degree $\delta \ge 2m+2 \ge 4$ satisfies $\lambda_2(G) < \delta - \frac{2m+1}{\delta+1}$, then $G$ contains at least $m+1$…

The largest order $n(d,k)$ of a graph of maximum degree $d$ and diameter $k$ cannot exceed the Moore bound, which has the form $M(d,k)=d^k - O(d^{k-1})$ for $d\to\infty$ and any fixed $k$. Known results in finite geometries on generalised…

We bound the second eigenvalue of random $d$-regular graphs, for a wide range of degrees $d$, using a novel approach based on Fourier analysis. Let $G_{n, d}$ be a uniform random $d$-regular graph on $n$ vertices, and let $\lambda (G_{n,…

Given a real $\mu\geq 1$, a graph $H$ is $\mu$-almost-regular if $\Delta(H)\leq \mu \delta(H)$. The celebrated regularization theorem of Erd\H{o}s and Simonovits states that for every real $0<\varepsilon<1$ there exists a real…

For a connected graph $G$ with order $n$ and an integer $k\geq 1$, we denote by $$S_k(D(G))=\lambda_1(D(G))+\cdots+\lambda_k(D(G))$$ the sum of $k$ largest distance eigenvalues of $G$. In this paper, we consider the sharp upper bound and…

A $k$-geodetic digraph with minimum out-degree $d$ has excess $\epsilon $ if it has order $M(d,k) + \epsilon $, where $M(d,k)$ represents the Moore bound for out-degree $d$ and diameter $k$. For given $\epsilon $, it is simple to show that…

A set of vertices $W$ resolves a graph $G$ if every vertex is uniquely determined by its vector of distances to the vertices in $W$. A metric dimension of $G$ is the minimum cardinality of a resolving set of $G$. A bipartite graph G(n,n) is…

In this paper we give graphs with the largest known order for a given degree $\Delta$ and diameter $D$. The graphs are constructed from Moore bipartite graphs by replacement of some vertices by adequate structures. The paper also contains…

Bollob\'{a}s and Scott [5] conjectured that every graph $G$ has a balanced bipartite spanning subgraph $H$ such that for each $v\in V(G)$, $d_H(v)\ge (d_G(v)-1)/2$. In this paper, we show that every graphic sequence has a realization for…

Let $G$ be a connected $d$-regular graph of order $n$, where $d\geq3$. Let $\lambda_{2}(G)$ be the second largest eigenvalue of $G$. For even $n$, we show that $G$ contains $\left\lfloor\frac{2}{3}(d-\lambda_{2}(G))\right\rfloor$…

The sum $\lambda_1 + \lambda_n$ of the maximum and minimum eigenvalues, and the odd girth of a graph both measure bipartiteness. We seek to relate these measures. In particular, for an odd integer $k\geq 3$, let $\gamma_k$ denote the…

Let $n(k, d)$ be the order of the largest undirected graphs of maximum degree $k$ and diameter $d$, and let $M(k,d)$ be the corresponding Moore bound. In this paper, we give a positive answer to the question of Bermond and Bollob\'as…

The degree-diameter problem asks for the maximum number of vertices in a graph with maximum degree $\Delta$ and diameter $k$. For fixed $k$, the answer is $\Theta(\Delta^k)$. We consider the degree-diameter problem for particular classes of…

Let $v(k, \lambda)$ be the maximum number of vertices of a connected $k$-regular graph with second largest eigenvalue at most $\lambda$. The Alon-Boppana Theorem implies that $v(k, \lambda)$ is finite when $k > \frac{\lambda^2 + 4}{4}$. In…

It is well-known that the Brualdi-Hoffman-Tur\'an-type problem inquiries about the maximum spectral radius \( \lambda(G) \) of an \( F \)-free graph \( G \) with \( m \) edges. Let \( \theta_{1,p,q} \) denote the theta graph, which is…

A $\frac{1}{k}$-majority $l$-edge-colouring of a graph $G$ is a colouring of its edges with $l$ colours such that for every colour $i$ and each vertex $v$ of $G$, at most $\frac{1}{k}$'th of the edges incident with $v$ have colour $i$. We…