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The index of a signed graph is the largest eigenvalue of its adjacency matrix. For positive integers $n$ and $m\le n^2/4$, we determine the maximal index of complete signed graphs with $n$ vertices and $m$ negative edges. This settles (the…

An almost self-centered graph is a connected graph of order $n$ with exactly $n-2$ central vertices, and an almost peripheral graph is a connected graph of order $n$ with exactly $n-1$ peripheral vertices. We determine (1) the maximum girth…

We improve the best known lower bounds on the exponential behavior of the maximum of the number of connected sets, $N(G)$, and dominating connected sets, $N_{dom}(G)$, for regular graphs. These lower bounds are improved by constructing a…

In this paper, we study the following question. Let $\mathcal G$ be a family of planar graphs and let $k\geq 3$ be an integer. What is the largest value $f_k(n)$ such that every $n$-vertex graph in $\mathcal G$ has an induced subgraph with…

We prove that the number of vertices of a polytope of a particular kind is exponentially large in the dimension of the polytope. As a corollary, we prove that an n-dimensional centrally symmetric polytope with O(n) facets has 2^{Omega(n)}…

We show that the median eigenvalues of every connected graph of maximum degree at most three, except for the Heawood graph, are at most $1$ in absolute value, resolving open problems posed by Fowler and Pisanski, and by Mohar.

This paper investigates the asymptotic nature of graph spectra when some edges of a graph are subdivided sufficiently many times. In the special case where all edges of a graph are subdivided, we find the exact limits of the $k$-th largest…

Let $G$ be a simple graph with $n$ vertices and $m$ edges and let $k$ be a natural number such that $k\leq n.$ Brouwer conjectured that the sum of the $k$ largest Laplacian eigenvalues of $G$ is at most $m+{k+1 \choose 2}.$ In this paper we…

The second-largest eigenvalue and second-smallest Laplacian eigenvalue of a graph are measures of its connectivity. These eigenvalues can be used to analyze the robustness, resilience, and synchronizability of networks, and are related to…

The problem of finding upper bounds for minimal vertex number of graphs with a given automorphism group is addressed in this article for the case of cyclic $2$-groups. We show that for any natural $n\ge 2$ there is an undirected graph…

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…

Let $G$ be a bipartite graph without loops and multiple edges on $v\ge 4$ vertices, which can be drawn on the plane such that any edge intersects at most one other edge. We prove that such graph has at most $3v-8$ edges for even $v\ne 6$…

In this paper, all graphs whose adjacency matrix has at most two eigenvalues (multiplicities included) different from $2$ and $-1$ are determined. These graphs conclude a class of generalized friendship graphs $F_{t,r,k}, $ which is the…

In this paper, we present an elementary proof of a theorem of Serre concerning the greatest eigenvalues of $k$-regular graphs. We also prove an analogue of Serre's theorem regarding the least eigenvalues of $k$-regular graphs: given…

We find the maximum number of maximal independent sets in two families of graphs: all graphs with $n$ vertices and at most $r$ cycles, and all such graphs that are also connected. In addition, we characterize the extremal graphs.

Given graphs H_1,...,H_k, we study the minimum order of a graph G such that for each i, the induced copies of H_i in G cover V(G). We prove a general upper bound of twice the sum of the numbers m_i, where m_i is one less than the order of…

The main purpose of this paper is to prove the uniqueness of a graph attaining the maximum of the number of independent sets over all $k$-regular graphs on $n$ vertices for $2k|n$.

Almost Moore mixed graphs\/} appear in the context of the degree/diameter problem as a class of extremal mixed graphs, in the sense that their order is one unit less than the Moore bound for such graphs. The problem of their existence has…

The spectrum of a graph is closely related to many graph parameters. In particular, the spectral gap of a regular graph which is the difference between its valency and second eigenvalue, is widely seen an algebraic measure of connectivity…

In this paper, we prove that for any $k\ge 3$, there exist infinitely many minimal asymmetric $k$-uniform hypergraphs. This is in a striking contrast to $k=2$, where it has been proved recently that there are exactly $18$ minimal asymmetric…