For a graph G, the irregularity and total irregularity of G are defined as irr(G)=∑_(uv∈E(G))〖|d_G (u)-d_G (v)|〗 and irr_t (G)=1/2 ∑_(u,v∈V(G))〖|d_G (u)-d_G (v)|〗, respectively, where d_G (u) is the degree of vertex u. In this paper, we characterize all ‎connected Eulerian graphs with the second minimum irregularity, the second and third minimum total irregularity value, respectively.

An undirected graph is Eulerian if it is connected and all its vertices are of even degree. Similarly, a directed graph is Eulerian, if for each vertex its in-degree is equal to its out-degree. It is well known that Eulerian graphs can be recognized in polynomial time while the problems of finding a maximum Eulerian subgraph or a maximum induced Eulerian subgraph are NP-hard. In this paper, we ...

In this paper, we show that if G is a 3-edge-connected graph with S V ðGÞ and jSj 12, then eitherG has an Eulerian subgraphH such that S V ðHÞ, or G can be contracted to the Petersen graph in such a way that the preimage of each vertex of the Petersen graph contains at least one vertex in S. If G is a 3-edge-connected planar graph, then for any jSj 23, G has an Eulerian subgraph H such that S V...

The Hamiltonian index ofa graph G is defined as h(G) = minim: LIIl(G) is Hamiltonian). In this paper, using the reduction method of Catlin [P.A. Catlin, A reduction method to find spanning Eulerian subgraphs, J. Graph Theory 12 (1988) 29-44], we constructed a graph i/(m)(G) from G and prove that ifh(G)?:. 2, then h(G) = minim : i/(m)(G) has a spanning Eulerian subgraphl.

Further work of Brylawski and Heron (see [4, p. 315]) explores other characterizations of Eulerian binary matroids. They showed, independently, that a binary matroid M is Eulerian if and only if its dual, M∗, is a binary affine matroid. More recently, Shikare and Raghunathan [5] have shown that a binary matroid M is Eulerian if and only if the number of independent sets of M is odd. This chapte...

Upper bounds on the maximum number of minimal codewords in a binary code follow from the theory of matroids. Random coding provide lower bounds. In this paper we compare these bounds with analogous bounds for the cycle code of graphs. This problem (in the graphic case) was considered in 1981 by Entringer and Slater who asked if a connected graph with p vertices and q edges can have only slightl...

A graph G is supereulerian if it has a spanning eulerian subgraph. We prove that every 3-edge-connected graph with the circumference at most 11 has a spanning eulerian subgraph if and only if it is not contractible to the Petersen graph. As applications, we determine collections F1, F2 and F3 of graphs to prove each of the following (i) Every 3-connected {K1,3, Z9}-free graph is hamiltonian if ...

In this paper, we study matching integral graphs of small order. A graph is called matching integral if the zeros of its matching polynomial are all integers. Matching integral graphs were first studied by Akbari, Khalashi, etc. They characterized all traceable graphs which are matching integral. They studied matching integral regular graphs. Furthermore, it has been shown that there is no matc...

Consider a search game with an immobile hider in a graph. A Chinese postman tour is a closed trajectory which visits all the points of the graph and has minimal length. We show that encircling the Chinese postman tour in a random direction is an optimal search strategy if and only if the graph is weakly Eulerian (i.e it consists of several Eulerian curves connected in a tree-like structure).