# On Strongly Regular Graphs with /m2 - m3/ â‰¤ 3

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Let D be a strongly connected digraph on n â‰¥ 4 vertices. A vertex v of D is noncritical if the digraph D âˆ’ v is strongly connected. It is proved that if the sum of degrees of any two adjacent vertices of D is at least n + 1, then there exists a noncritical vertex in D, and if the...

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For vertices x and y in a connected graph G, the detour distance D (x, y) is the length of a longest x-y path in G. An x-y path of length D (x, y) is an x-y detour. The closed detour interval ID[x, y] consists of x, y, and all vertices lying on some x-y detour of G; while for S âŠ† V (G),...

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For a spanning tree T of a connected graph .. and for a labelling Î¦: E(T ) ! {+,-}, Î¦ is called an alternating sign on a spanning tree T of a graph ... if for any cotree edge e âˆˆ E(...)-E(T ), the unique path in T joining both end vertices of e has alternating signs. In the present...

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For an integer $$k$$ with $$k \ge 2$$ , a $$k$$ - tree (resp. a $$k$$ - forest) is a tree (resp. forest) with maximum degree at most $$k$$ . In this paper, we show that for any integer $$k$$ with $$k \ge 3$$ , any connected $$K_{1,k+1}$$ -free graph has a spanning $$k$$ -tree or a spanning $$k$$...

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The existence problem on trees preserving the Wiener index of two classes of graphs is studied in this paper. The Wiener index W(G) of a connected graph G is the sum of distances among all pairs of vertices in G. If there is a connected subtree T of a given connected graph G such that W(G)...

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Given a finite nonempty set of primes S, we build a graph $${\mathcal{G}}$$ with vertex set $${\mathbb{Q}}$$ by connecting $${x, y \in \mathbb{Q}}$$ if the prime divisors of both the numerator and denominator of x âˆ’ y are from S. In this paper we resolve two conjectures posed by Ruzsa...

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In this paper necessary and sufficient conditions are found for the existence of a $$4$$ -cycle system of a complete graph on $$n$$ vertices with leave a nearly $$2$$ -regular graph (that is, a not necessarily spanning graph in which all vertices have degree 2 except for one of degree greater...

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A degree-constrained graph orientation of an undirected graph G is an assignment of a direction to each edge in G such that the outdegree of every vertex in the resulting directed graph satisfies a specified lower and/or upper bound. Such graph orientations have been studied for a long time and...

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An important property of a scale-free network which allows one to communicate between its nodes is the connectivity. From the simulation results it is known that such networks are resistant to random damage. It is also known from experiments that usually disconnecting such a graph is obtained by...