Related papers: A partial proof of the Brouwer's conjecture

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…

Nikiforov conjectured that for a given integer $k\ge 2$, any graph $G$ of sufficiently large order $n$ with spectral radius $\mu(G)\geq \mu(S_{n,k})$ (or $\mu(G)\ge \mu(S_{n,k}^+))$ contains $C_{2k+1}$ or $C_{2k+2}$(or $C_{2k+2}$), unless…

Let $G$ be a simple graph. A pendant path of $G$ is a path such that one of its end vertices has degree $1$, the other end has degree $\ge3$, and all the internal vertices have degree $2$. Let $p_k(G)$ be the number of pendant paths of…

For given integers $k$ and $\ell$ with $0<\ell< {k \choose 2}$, Alon, Hefetz, Krivelevich and Tyomkyn formulated the following conjecture: When sampling a $k$-vertex subset uniformly at random from a very large graph $G$, then the…

For a simple graph $G$, the energy $\mathcal{E}(G)$ is defined as the sum of the absolute values of all the eigenvalues of its adjacency matrix $A(G)$. Let $n, m$, respectively, be the number of vertices and edges of $G$. One well-known…

Let $(n^+, n^0, n^-)$ denote the inertia of a graph $G$ with $n$ vertices. Nordhaus-Gaddum bounds are known for inertia, except for an upper bound for $n^-$. We conjecture that for any graph \[ n^-(G) + n^-(\bar{G}) \le 1.5(n - 1), \] and…

Denote by $q_n(G)$ the smallest eigenvalue of the signless Laplacian matrix of an $n$-vertex graph $G$. Brandt conjectured in 1997 that for regular triangle-free graphs $q_n(G) \leq \frac{4n}{25}$. We prove a stronger result: If $G$ is a…

Mader proved that for $k\geq 2$ and $n\geq 2k$, every $n$-vertex graph with no $(k+1)$-connected subgraphs has at most $(1+\frac{1}{\sqrt{2}})k(n-k)$ edges. He also conjectured that for $n$ large with respect to $k$, every such graph has at…

Let $G$ be a graph with $p(G)$ pendant vertices and $q(G)$ quasi-pendant vertices. Denote by $m_{L(G)}(\lambda)$ the multiplicity of $\lambda$ as a Laplacian eigenvalue of $G$. Let $\overline{G}$ be the reduced graph of $G$, which can be…

Let $f(n,k)$ be the minimum number of edges that must be removed from some complete geometric graph $G$ on $n$ points, so that there exists a tree on $k$ vertices that is no longer a planar subgraph of $G$. In this paper we show that…

The booksize $ \mathrm{bk}(G) $ of a graph $ G $, introduced by Erd\H{o}s, refers to the maximum integer $ r $ for which $G$ contains the book $ B_r $ as a subgraph. This paper investigates two open problems in spectral graph theory related…

An equitable $k$-coloring of a graph is a proper $k$-coloring where the sizes of any two different color classes differ by at most one. In 1973, Meyer conjectured that every connected graph $G$ has an equitable $k$-coloring for some $k\leq…

Let $\alpha'$ and $\mu_i$ denote the matching number of a non-empty simple graph $G$ with $n$ vertices and the $i$-th smallest eigenvalue of its Laplacian matrix, respectively. In this paper, we prove a tight lower bound $$\alpha' \ge…

Let $\lambda_1(G)\ge \lambda_2(G)\ge \cdots \ge \lambda_n(G)$ denote the adjacency eigenvalues of a graph $G$ of order $n$. We prove that for every $k\geq 2$ and every graph $G$ on $n\geq k$ vertices, $$ \lambda_k(G)\le…

Let $q_{\min}(G)$ stand for the smallest eigenvalue of the signless Laplacian of a graph $G$ of order $n.$ This paper gives some results on the following extremal problem: How large can $q_\min\left( G\right) $ be if $G$ is a graph of order…

Given a connected graph $G$, the Randi\'c index $R(G)$ is the sum of $\tfrac{1}{\sqrt{d(u)d(v)}}$ over all edges $\{u,v\}$ of $G$, where $d(u)$ and $d(v)$ are the degree of vertices $u$ and $v$ respectively. Let $q(G)$ be the largest…

We prove that the maximum eigenvalue of the (both signed and unsigned) Laplacian of level $k$ Kikuchi graph of any graph $G$ with $m$ edges is at most $m+k$. This confirms four recent conjectures of Apte, Parekh, and Sud. As applications,…

Let k_r(n,m) denote the minimum number of r-cliques in graphs with n vertices and m edges. For r=3,4 we give a lower bound on k_r(n,m) that approximates k_r(n,m) with an error smaller than n^r/(n^2-2m). The solution is based on a constraint…

We give a new unified proof that any simple graph on $n$ vertices with maximum degree at most $\Delta$ has no more than $a\binom{\Delta+1}{t}+\binom{b}{t}$ cliques of size $t \ (t \ge 3)$, where $n = a(\Delta+1)+b \ (0 \le b \le \Delta)$.

A graph $G$ is $k$-{\em critical} if it has chromatic number $k$, but every proper subgraph of $G$ is $(k-1)$--colorable. Let $f_k(n)$ denote the minimum number of edges in an $n$-vertex $k$-critical graph. In a very recent paper, we gave a…