Related papers: A note on Fiedler value of classes with sublinear …

Let $\lambda_{2}(G)$ be the second smallest normalized Laplacian eigenvalue of a graph $G$. In this paper, we determine all unicyclic graphs of order $n\geq21$ with $\lambda_{2}(G)\geq 1-\frac{\sqrt{6}}{3}$. Moreover, the unicyclic graphs…

The zero locus of a function f on a graph G is defined as the graph with vertex set consisting of all complete subgraphs of G, on which f changes sign and where x,y are connected if one is contained in the other. For d-graphs, finite simple…

Let $G$ be a simple connected graph with vertex set $V(G)=\{v_{1}, v_{2}, \ldots, v_{n}\}$. The distance $d_G(v_i,v_j)$ between two vertices $v_i$ and $v_j$ of $G$ is the length of a shortest path between $v_i$ and $v_j$. The distance…

An identifying code of a graph G is a dominating set C such that every vertex x of G is distinguished from all other vertices by the set of vertices in C that are at distance at most 1 from x. The problem of finding an identifying code of…

Rigid frameworks in some Euclidian space are embedded graphs having a unique local realization (up to Euclidian motions) for the given edge lengths, although globally they may have several. We study the number of distinct planar embeddings…

The speed of a class of graphs counts the number of graphs on the vertex set $\lbrace 1,\dots, n\rbrace$ inside the class as a function of $n$. In this paper, we investigate this function for many classes of graphs that naturally arise in…

For a connected graph $G$, let $A(G)$ be the adjacency matrix of $G$ and $D(G)$ be the diagonal matrix of the degrees of the vertices in $G$. The $A_{\alpha}$-matrix of $G$ is defined as \begin{align*} A_\alpha (G) = \alpha D(G) +…

A pseudo [2,b]-factor of a graph G is a spanning subgraph in which each component C on at least three vertices is a [2,b]-graph. The main contibution of this paper, is to give an upper bound to the number of components that are edges or…

Let $G$ be a graph on $n$ vertices of independence number $\alpha(G)$ such that every induced subgraph of $G$ on $n-k$ vertices has an independent set of size at least $\alpha(G) - \ell$. What is the largest possible $\alpha(G)$ in terms of…

In this paper, some subclasses of block graphs are considered in order to analyze Fiedler vector of its members. Two families of block graphs with cliques of fixed size, the block-path and block-starlike graphs, are introduced. Cases A and…

Given a family ${\cal F}$ of graphs, and a positive integer $n$, the Tur\'an number $ex(n,{\cal F})$ of ${\cal F}$ is the maximum number of edges in an $n$-vertex graph that does not contain any member of ${\cal F}$ as a subgraph. The order…

For a graph $G$ and a positive integer $c$, let $M_c(G)$ be the size of a subgraph of $G$ induced by a randomly sampled subset of $c$ vertices. Second-order moments of $M_c(G)$ encode part of the structure of $G$. We use this fact, coupled…

We say that a first order formula A distinguishes a graph G from another graph G' if A is true on G and false on G'. Provided G and G' are non-isomorphic, let D(G,G') denote the minimal quantifier rank of a such formula. We prove that, if G…

Given a simple graph $G$, its $A_\alpha$ matrix is a convex combination with parameter $\alpha\in [0,1]$ of its adjacency matrix and its degree diagonal matrices. Here we compare two lower bounds presented in [J. D. G. Silva Jr., C. S.…

We investigate the product structure of hereditary graph classes admitting strongly sublinear separators. We characterise such classes as subgraphs of the strong product of a star and a complete graph of strongly sublinear size. In a more…

Given an $n$-vertex graph $G$, let $\hom (G)$ denote the size of a largest homogeneous set in $G$ and let $f(G)$ denote the maximal number of distinct degrees appearing in an induced subgraph of $G$. The relationship between these…

In this paper, we completely characterize the graphs with third largest distance eigenvalue at most $-1$ and smallest distance eigenvalue at least $-3$. In particular, we determine all graphs whose distance matrices have exactly two…

The Euclidean dimension a graph $G$ is defined to be the smallest integer $d$ such that the vertices of $G$ can be located in $\mathbb{R}^d$ in such a way that two vertices are unit distance apart if and only if they are adjacent in $G$. In…

Wiener index, defined as the sum of distances between all unordered pairs of vertices, is one of the most popular molecular descriptors. It is well known that among 2-vertex connected graphs on $n\ge 3$ vertices, the cycle $C_n$ attains the…

We obtain a second order differential equation on moduli space satisfied by certain modular graph functions at genus two, each of which has two links. This eigenvalue equation is obtained by analyzing the variations of these graphs under…