{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:56Z","timestamp":1753893836918,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>The Moore bound $M(k,g)$ is a lower bound on the\u00a0order of $k$-regular graphs of girth $g$ (denoted $(k,g)$-graphs). The excess $e$ of a $(k,g)$-graph of order $n$ is the difference $ n-M(k,g) $. In this paper we consider the existence of $(k,g)$-bipartite graphs of excess $4$ by studying spectral properties of their adjacency matrices. For a given graph $G$ and for the integers $i$ with $0\\leq i\\leq diam(G)$, the $i$-distance matrix $A_i$ of $G$ is an $n\\times n$ matrix\u00a0such that the entry in position $(u,v)$ is $1$ if the distance between the vertices $u$ and $v$ is $i$, and zero otherwise.\u00a0We prove that the $(k,g)$-bipartite graphs of excess $4$ satisfy the equation $kJ=(A+kI)(H_{d-1}(A)+E)$, where $A=A_{1}$ denotes the adjacency matrix of the graph in question, $J$ the $n \\times n$ all-ones matrix, $E=A_{d+1}$ the adjacency matrix of a union of\u00a0vertex-disjoint cycles, and $H_{d-1}(x)$ is the Dickson polynomial of the second kind with parameter $k-1$ and degree $d-1$.\u00a0We observe that the eigenvalues other than $\\pm k$ of these graphs are roots of the polynomials $H_{d-1}(x)+\\lambda$, where $\\lambda$ is an eigenvalue of $E$. Based on the irreducibility of $H_{d-1}(x)\\pm2$, we give necessary conditions for the existence of these graphs.\u00a0If $E$ is the adjacency matrix of a cycle of order $n$, we call the corresponding graphs graphs with cyclic excess; if $E$ is the adjacency matrix of a disjoint union of two cycles, we call the corresponding graphs graphs with bicyclic excess. In this paper we prove the non-existence of $(k,g)$-graphs with cyclic excess $4$ if $k\\geq6$ and $k \\equiv1 \\!\\! \\pmod {3}$, $g=8, 12, 16$ or $k \\equiv2 \\!\\! \\pmod {3}$, $g=8;$ and the non-existence of $(k,g)$-graphs with bicyclic excess $4$ if $k\\geq7$ is an odd number and $g=2d$ such that $d\\geq4$ is even.<\/jats:p>","DOI":"10.37236\/6693","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T19:10:27Z","timestamp":1578683427000},"source":"Crossref","is-referenced-by-count":3,"title":["On Bipartite Cages of Excess 4"],"prefix":"10.37236","volume":"24","author":[{"given":"Slobodan","family":"Filipovski","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2017,3,3]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v24i1p40\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v24i1p40\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T05:03:42Z","timestamp":1579237422000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v24i1p40"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,3,3]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2017,1,20]]}},"URL":"https:\/\/doi.org\/10.37236\/6693","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2017,3,3]]},"article-number":"P1.40"}}