Related papers: On split graphs with four distinct eigenvalues
Taking the d-th distance power of a graph, one adds edges between all pairs of vertices of that graph whose distance is at most d. It is shown that only the numbers -3, -2, -1, 0, 1, 2d can be integer eigenvalues of a circuit distance…
Let ${\rm ind}(G)$ be the number of independent sets in a graph $G$. We show that if $G$ has maximum degree at most $5$ then $$ {\rm ind}(G) \leq 2^{{\rm iso}(G)} \prod_{uv \in E(G)} {\rm ind}(K_{d(u),d(v)})^{\frac{1}{d(u)d(v)}} $$ (where…
This paper considers the degree-diameter problem for extremal and largest known undirected circulant graphs of degree 2 to 9 of arbitrary diameter. As these graphs are vertex transitive it is possible to define their distance partition. The…
The minimum number of distinct eigenvalues, taken over all real symmetric matrices compatible with a given graph $G$, is denoted by $q(G)$. Using other parameters related to $G$, bounds for $q(G)$ are proven and then applied to deduce…
We consider the bipartite version of the {\it degree/diameter problem}, namely, given natural numbers $d\ge2$ and $D\ge2$, find the maximum number $\N^b(d,D)$ of vertices in a bipartite graph of maximum degree $d$ and diameter $D$. In this…
For a given positive integer t we consider graphs having maximal independent sets of precisely t distinct cardinalities and restrict our attention to those that have no vertices of degree one. In the situation when t is four or larger and…
The \emph{segment number} of a planar graph is the smallest number of line segments whose union represents a crossing-free straight-line drawing of the given graph in the plane. The segment number is a measure for the visual complexity of a…
Sharp bounds on the least eigenvalue of an arbitrary graph are presented. Necessary and sufficient (just sufficient) conditions for the lower (upper) bound to be attained are deduced using edge clique partitions. As an application, we prove…
The degree-based entropy of a graph is defined as the Shannon entropy based on the information functional that associates the vertices of the graph with the corresponding degrees. In this paper, we study extremal problems of finding the…
A $k$-regular graph of girth $g$ is called edge-girth-regular graph, shortly egr-graph, if each of its edges is contained in exactly $\lambda$ distinct $g-$cycles. An egr-graph is called extremal for the triple $(k, g, \lambda)$ if has the…
In this paper, we consider the distance spectra of the derangement graphs. First we give a constructive proof that the connected derangement graphs are of diameter 2. Then we obtain their distance spectra. In particular, we determine all…
Define a boundary point of a graph which is embedded in the Euclidean plane a vertex which is incident to only one edge. In this paper we consider graphs which are embedded in the Euclidean plane with a finite number of boundary points. The…
An extremal graph for a graph $H$ on $n$ vertices is a graph on $n$ vertices with maximum number of edges that does not contain $H$ as a subgraph. Let $T_{n,r}$ be the Tur\'{a}n graph, which is the complete $r$-partite graph on $n$ vertices…
A graph is split if there is a partition of its vertex set into a clique and an independent set. The present paper is devoted to the splitness of some graphs related to finite simple groups, namely, prime graphs and solvable graphs, and…
Let $D$ be a strongly connected digraph. An arc set $S$ of $D$ is a \emph{restricted arc-cut} of $D$ if $D-S$ has a non-trivial strong component $D_{1}$ such that $D-V(D_{1})$ contains an arc. The \emph{restricted arc-connectivity}…
We consider numbers and sizes of independent sets in graphs with minimum degree at least $d$, when the number $n$ of vertices is large. In particular we investigate which of these graphs yield the maximum numbers of independent sets of…
We investigate graph based secret sharing schemes and its information ratio, also called complexity, measuring the maximal amount of information the vertices has to store. It was conjectured that in large girth graphs, where the interaction…
The degree diameter problem asks for the maximum possible number of vertices in a graph of maximum degree $\Delta$ and diameter $D$. In this paper, we focus on planar graphs of diameter $3$. Fellows, Hell and Seyffarth (1995) proved that…
Motivated by applications in graph drawing and information visualization, we examine the planar split thickness of a graph, that is, the smallest $k$ such that the graph is $k$-splittable into a planar graph. A $k$-split operation…
This paper considers the degree-diameter problem for undirected circulant graphs. The focus is on extremal graphs of given (small) degree and arbitrary diameter. The published literature only covers graphs of up to degree 7. The approach…